A representation theoretic explanation of the Borcea–Brändén characterization

In 2009, Borcea and Brändén characterized all linear operators on multivariate polynomials which preserve the property of being non-vanishing (stable) on products of prescribed open circular regions. We give a representation theoretic interpretation of their findings, which generalizes and simplifies their result and leads to a conceptual unification of many related results in polynomial stability theory. At the heart of this unification is a generalized Grace’s theorem which addresses polynomials whose roots are all contained in some real interval or ray. This generalization allows us to extend the Borcea–Brändén result to characterize a certain subclass of the linear operators which preserve such polynomials.


Introduction
In 1914, Pólya and Schur [14] characterized the set of diagonal linear operators on polynomials which preserve real-rootedness. Since this seminal paper, much work has been done in extending this characterization to other classes of linear operators. This program in essence came to a close in 2009 with a paper of Borcea and Brändén [1], which gave a complete characterization of linear operators on polynomials which preserve real-rootedness.
Their real-rootedness preservation characterization is derived from a more general result pertaining to stable polynomials. Given ⊂ C m , we say that a polynomial f ∈ C[x 1 , . . . , x m ] is -stable if f does not vanish in . Further, f is real stable if it has real coefficients and is H m + -stable, where H + ⊂ C is the open upper half-plane. (We also denote the open lower half-plane by H − ⊂ C.) We additionally use the terms weaklystable and weakly real stable if we allow f ≡ 0. Finally, we write f ∈ C λ [x 1 , . . . , x m ] for λ ≡ (λ 1 , . . . , λ m ) if f is of degree at most λ k in x k . We are then led to the following problems for K ∈ {C, R}, generalized from the Pólya-Schur characterization: In [1], Borcea and Brändén were able to solve these problems in many cases. In particular, they solved both problems for K = R and = H m + , where m = 1 corresponds to the case of preservation of real-rooted polynomials. For K = C, they were able to solve Problem 1 for that is any product of open circular regions in C.
In this paper we will only be concerned with Problem 1, for which we now state the solution from [1]. Given a linear operator T : K λ [x 1 , . . . , x m ] → K[x 1 , . . . , x m ], a polynomial Symb B B (T ) called the (Borcea-Brändén) symbol is associated to T . Specifically, the symbol is a polynomial in K λ λ [x 1 , . . . , x m , z 1 , . . . , z m ] (i.e., of 2m variables), where λ λ := (λ 1 , . . . , λ m , λ 1 , . . . , λ m ). The crucial feature of the symbol is that it shares certain stability properties with its associated linear operator, which yields the characterizations stated in the following results. (We will express these results in more detail in Sects. 6 and 7.) To deal with other products of circular regions, one then conjugates T by certain Möbius transformations and applies Theorem 1.1 to the conjugated operator. Unfortunately though, this is a tedious process which has to be done each time a new stability region is to be considered. Additionally, the image dimension restrictions give rise to degeneracy cases which have to be dealt with separately. Both of these issues obscure the connection between an operator and its symbol.
In this paper, we present a new conceptual approach to the Borcea-Brändén characterization via the representation theory of SL 2 (C). In particular, we derive a new symbol (denoted Symb) in a natural way, and our definition eliminates the issues discussed above. This is seen in the following results, which are our simplified and generalized versions of the Borcea-Brändén characterizations. Note that for the sake of simplicity, we have omitted a few details here regarding non-convex circular regions. Specifically, circular regions of C should be thought of as lying in the Riemann sphere, so that complements of discs contain the point at ∞. Theorem 6.2 Fix a linear operator T : C λ [x 1 , . . . , x m ] → C α [x 1 , . . . , x l ], a product of all open or all closed circular regions 0 = C 1 × · · · × C m , and a product of sets 1 := S 1 × · · · × S m . Further, denote 0 := (C\C 1 ) × · · · × (C\C m ). Up to certain degree and convexity (of C k ) restrictions, we have that T maps 0 -stable polynomials to nonzero 1 We summarize the specific improvements that this and our other related results give over the Borcea-Brändén characterization as follows. 1. Different stability regions can be considered using the same symbol. The symbol we define in this paper is universal: for example, it gives stability-preservation information for any product of open circular regions. The Borcea-Brändén symbol, on the other hand, required the application of Möbius transformations. In addition, our symbol also allows for the output stability region to be chosen independently of the input stability region. While this does not literally improve the result, it does allow for quicker computations.
In particular, see Examples 6.3 and 6.4 where classical polynomial convolution results are easily derived from our framework. 2. Our characterization does not require any degeneracy condition. Our results characterize operators which preserve (strong) stability rather than weak stability. As seen above, this slightly stronger notion of stability enables us to eliminate any image dimension degeneracy condition, as required in the Borcea-Brändén characterizations (Theorems 1.1 and 1.2). This demonstrates a cleaner link between an operator and its symbol. 3. Closed circular regions and projectively convex regions can be considered. The symbol we define in this paper handles products of open circular regions, as well as products of closed circular regions. (In [12], Melamud proves a result similar to the Borcea-Brändén characterization for closed circular regions.) Further, we are also able to consider more general projectively convex regions (circular regions with portions of their boundary; also called generalized circular regions, see [18] and [17]) in Proposition 6.5. This allows us to determine stability-preservation information about real intervals and half-lines. It also turns out, somewhat surprisingly, that our symbol can handle products of any sets as possible output space stability regions (as seen in Theorem 6.2 above).
In the process of generalizing the Borcea-Brändén characterization we develop a general algebraic framework which also encompasses many of the classical polynomial tools. This framework aims to motivate classical results and provide intuition for the connection between a stability preserving operator and its symbol.

The main idea
A major purpose of this paper is to explain a certain conceptual thread in the history of polynomial stability theory: that it is often possible to determine general stability information from restricted sets of polynomials. For example, the Pólya-Schur and Borcea-Brändén characterizations derive from a single polynomial (i.e., the symbol) stability properties of a whole collection of polynomials in the output of a given linear operator. Additionally, the Grace-Walsh-Szegő coincidence theorem says that stability information of any polynomial can be determined from its polarization, which is of degree at most one in each variable.
As it turns out, these sorts of phenomena can be explained using relatively basic algebraic and representation theoretic concepts. We view C n [x] as a representation of SL 2 (C) via the standard action, given as follows. For φ ∈ SL 2 (C) and f ∈ C n [x], we define: Here φ −1 acts on x ∈ C as a Möbius transformation, or equivalently φ acts on the roots of f . (Similarly, C λ [x 1 , . . . , x m ] can be viewed as a representation of (SL 2 (C)) m via this action in each variable.) Under this interpretation, important maps like polarization, projection, the apolarity form, and even the symbol turn out to be invariant under these SL 2 (C) actions. This leads us to a conceptual thesis: SL 2 (C)-invariant maps transfer stability information.
The goal of this paper is then to explicate and answer the most important question related to this thesis: what does it mean for the symbol map to be SL 2 (C)-invariant and how does it transfer stability information? To answer this, we consider the following standard ideas relating spaces of linear operators to tensor products.
Let W 1 , W 2 be two finite dimensional representations of a group G, and let Hom(W 1 , W 2 ) denote the space of linear maps from W 1 to W 2 . Then, Hom(W 1 , W 2 ) ∼ = W * 1 W 2 (the outer tensor product) can viewed as a representation of G × G. If we further have a G-invariant bilinear form on W 1 , then we also have W 1 ∼ = W * 1 . This leads to the following identification: If W 1 and W 2 are spaces of polynomials, each in m variables, then their tensor product W 1 W 2 is isomorphic to a larger space of polynomials in 2m variables. That is, a linear operator between polynomial spaces W 1 and W 2 can be associated to some polynomial in double the variables, via the above identification of representations. This is precisely the idea of the symbol of an operator.
Let's see how this works in the univariate case. Consider C n [x] as a representation of the group SL 2 (C), as described above. It is then a standard result that the classical bilinear apolarity form is invariant under the action of Möbius transformations. That is, the apolarity form is an SL 2 (C)-invariant bilinear form on C n [x]. This form, applied to f , g ∈ C n [x] with coefficients f k , g k , is defined as follows: With this, we obtain the identification described above: The final piece of the puzzle is then to find a way to transfer stability information through this identification of representations. The key result to this end is the classical Grace's theorem: be polynomials of degree exactly n. Further, let C be some open or closed circular region such that f is C-stable and g is (C\C)-stable. Then, f , g n = 0.
That is, the apolarity form not only provides the link between a linear operator and its symbol, but also captures stability information. So, whatever stability claims we can make about polynomials in C (n,m) [x, z] can then be seamlessly transferred to corresponding linear operators in Hom(C n [x], C m [x]). From this we are able to recover the Borcea-Brändén characterization. Additionally, all of the theory here relating stability and the representation theory of SL 2 (C) can be generalized to multivariate polynomials in a straightforward manner. The details will be discussed in Sect. 3.
In a similar fashion, other important maps also have SL 2 (C)-invariance properties (e.g., polarization and projection, as used in the Grace-Walsh-Szegő coincidence theorem, explicitly give the isomorphisms of a classical representation theoretic result; see Appendix B). A main feature of our conceptual thesis is that it allows for a unification of many seemingly related results in polynomial stability theory. A crucial point to make then is that Grace's theorem is at the heart of this unification. That said, a significant portion of this paper is devoted to discussing it.

A generalized Grace's theorem and interval-and ray-rootedness
In [2], Borcea and Brändén are able to prove a multivariate Grace's theorem using their operator characterization. In this paper we will prove the multivariate version from scratch, and then use it to derive a new characterization of stability-preserving linear operators. In addition, we generalize it to projectively convex regions, which consist of an open circular region with a portion of its boundary (see Sect. 4.2). We state our new result as follows.
Note that this result can be seen as an extension of the generalized Grace's theorem given in Corollary 4.4 of [17].
x m ] such that f and g both have a nonzero term of degree λ. Also, denote C := H + ∪ R + and C := H − ∪ R − . If f is C m -stable and g is C m -stable, then f , g λ = 0.
This result can, for instance, give stability information about positive-and negative-rooted polynomials. Since the apolarity form is invariant under the action of Möbius transformations, we immediately obtain similar statements regarding the union of any open circular region and a portion of its boundary. Notice also that, unlike the classical Grace's theorem, the stability regions C and C have non-empty intersection.
In the vein of this extension, we provide a new characterization of a certain class of linear operators which preserve ray-and interval-rootedness. The problem of classifying all such operators is still open in general (see e.g., the end of [3]). Here, we solve this problem for a restricted class of operators: namely, operators which both preserve weak real-rootedness and also preserve ray-or interval-rootedness. Our main result in this direction is stated as follows, where a polynomial is called J -rooted when all of its roots are in J : In Sect. 7.4, this result is stated in a more restricted manner as the degeneracy condition (image dimension) and degree restrictions end up being more tedious than in the other results. Corollary 7.9 and further explication then give the result as stated here.
As a final note, all of the results given here in the introduction are stated slightly differently in Sects. 5, 6, 7. In particular, the notation V (λ) is used in place of C λ [x 1 , . . . , x m ], and reference is made to CP 1 (i.e., the Riemann sphere). This notation has to do with consideration of "roots at infinity", which allows us to remove degree restrictions and avoid reference to convex circular regions. We discuss this rigorously in Sect. 2.2.

A roadmap
We now describe the content of the remainder of this paper. In Sect. 2, we discuss the use of homogeneous polynomials via the notation V (n) and V (λ), and we describe the relation of these spaces to the notion of roots at infinity.
In Sect. 3, we explicate some very basic representation theory of SL 2 (C). We then demonstrate how the apolarity form and the symbol arise as natural constructs in this context. Results like the symbol lemma (Lemma 3.7) and the SL 2 (C)-invariance of the apolarity form are stated here.
In Sect. 4, we discuss some classical and some new polynomial stability theory results, and their multivariate analogues, in the homogeneous context. We also extend Laguerre's theorem to projectively convex regions (generalized circular regions), which later allows us to prove results regarding polynomials which have all their roots in a given interval.
In Sect. 5, we state and prove our generalized Grace's theorem. We also discuss other stability regions to which the theorem applies, and consider symbols of linear operators given by evaluation at a particular point. We call these polynomials evaluation symbols, as they turn out to play a crucial role in the proofs of the operator characterizations.
In Sect. 6, we finally state and prove our improved characterizations of stability-preserving operators. We also demonstrate how the Borcea-Brändén characterizations can be seen (with a bit of work) to be corollaries of our characterizations. We then provide examples of the use of our results. In particular, we show how stability results related to classical polynomial convolutions can be immediately recovered.
In Sect. 7, we state and prove the analogous characterization of strong real stabilitypreserving operators. As with complex operators, we show how the Borcea-Brändén characterization can be obtained as a corollary. In this section, we also state and prove our characterization of operators which preserve both weak real-rootedness as well as interval-(or ray-) rootedness.
In Appendix A, we explicate more of the representation theory of SL 2 (C) in a polynomialminded way. Specifically, we prove a few standard tensor product decomposition results which more clearly demonstrate how this theory connects to the notion of apolarity.
In Appendix B, we discuss how polarization and the Grace-Walsh-Szegő coincidence theorem fit in to the framework presented in this paper. We also demonstrate that the classical isomorphism V (n) ∼ = Sym n (V (1)) (for representations of SL 2 (C)) can be realized as the polarization map and therefore has a stability-theoretic interpretation. While important to the conceptual thesis stated above, we place this discussion in an appendix as it is not utilized elsewhere.

Preliminaries
Here, we discuss basic notation and results related to polynomials and stability. In particular, we discuss in more detail the notation and consequences related to the use of homogeneous polynomials in place of usual univariate and mutlivariate polynomials. Then, we state a number of basic stability results in the language of homogeneous polynomials.

Notation
x m ] are of degree at most λ k in the variable x k . In particular, we call polynomials in C (1 m ) [x 1 , . . . , x m ] multi-affine. We will also use the shorthand C n [x] to refer to univariate polynomials of degree at most n. Now we define similar spaces of polynomials which are homogeneous in pairs of variables. These polynomials should be seen as per-variable homogenizations of polynomials of the spaces defined above. For λ = (λ 1 , . . . , λ m ) ∈ N m 0 and K = C or K = R, we define: We also use the shorthand V (λ) = V C (λ). As above, we call polynomials in V (1 m ) multiaffine. The notation used here is generalized from what is typically used to denote the irreducible representations of SL 2 (C). As it turns out, spaces of homogeneous polynomials in two variables can be used to define these representations. This will be made more precise in Sect. 3.
We let CP 1 denote the projective space of lines in C 2 , and we will also identify this space with the Riemann sphere. Note that CP 1 can also be considered as a compact version of C with one extra point added at infinity. We will often identify CP 1 with C (up to this extra point) via stereographic projection. A circular region is then an open or closed disc, halfplane, or complement of a disc in C. The set of circular regions is transitive under the action of Möbius transformations. We also use the name circular regions to refer to the stereographic projections into CP 1 . Closed half-planes and complements of discs projected into CP 1 will contain the point at infinity. Throughout, we will let ∂ S denote the boundary of (the closure of) the set S in CP 1 and let S • denote the interior of S.

Homogeneous polynomials
The usual degree-n homogenization of a polynomial f ∈ C n [x] is defined on monomials as follows and is extended linearly.
Hmg n : C n [x] → V (n) x k → x k y n−k More generally, for λ ∈ N m 0 the degree-λ homogenization is defined on monomials as follows and is extended linearly.
and V (n) are isomorphic as vector spaces via Hmg n , and we will mainly utilize bivariate homogeneous polynomials in V (n) over the usual univariate polynomials in C n [x]. What homogeneity gets us is a simplification of a number of issues related to the fact that polynomials in C n [x] have at most n zeros. Specifically, it is more natural to think of the missing zeros (when the number of zeros is less than n) as being "at infinity". Certain results require premises restricting to convex regions or to polynomials of degree exactly n (e.g., the classical Grace-Walsh-Szegő coincidence theorem), and such details vanish when considering homogeneous polynomials with possible roots at infinity. Another way to say this is that we consider polynomials in V (n) to have exactly n roots in CP 1 , which can also be thought of as the Riemann sphere. We also consider polynomials p(x, y) = p(x 1 , y 1 , . . . , x m , y m ) ∈ V (λ) to have zeros in (CP 1 ) m , where each pair (x k , y k ) corresponds to a single factor of CP 1 in (CP 1 ) m .
We use the notation (a : b) ∈ CP 1 , which is meant to give off the connotation of a ratio; that is, (a : b) should feel like a/b. We also use the notation (a : b) = (a 1 : b 1 ), . . . , (a m : b m ) ∈ (CP 1 ) m . Note that this connotation aligns with the idea of considering the zeros of polynomials to be in (CP 1 ) m . given the following equality. Defining p := Hmg λ ( f ) for a given polynomial f ∈ C λ [x 1 , . . . , x m ], we have: Finally, we give an important definition which will essentially replace the notion of a monic polynomial for homogeneous polynomials.
Definition 2.1 Given p ∈ V (λ), we say that p is top-degree monic if the coefficient of x λ in p equals 1. In particular, if p ∈ V (n) is top-degree monic, then p has no roots at infinity.

Homogeneous polynomials as representations
In this section, we will discuss some basic representation theory of SL 2 (C) and show how the apolarity form and the notion of the symbol of an operator arise naturally in the representation theoretic context. Most of the representation theory we use in this section is very basic. There are a number of references which discuss the theory in full detail, albeit with different goals in mind. Typically this is done via the theory of Lie groups and algebras, as in [6] and in [10]. As a note, most of the content of this section is less relevant to the analytic questions associated to polynomials. Rather, it serves as the foundational structure for a new approach to Grace's theorem and results concerning stability-preserving operators. For this reason, we believe it worthwhile to explicate key aspects of this foundation and their connection to analytic results. Pushing further into this connection may lead to new results beyond the scope of this paper.
This then induces an action on V (n) by acting on the roots (in CP 1 ) of polynomials in V (n). Given p ∈ V (n) and φ ∈ SL 2 (C), this action is defined via: We can define a similar action of (SL 2 (C)) m on V (λ), for λ ∈ N m 0 . Specifically, given p ∈ V (λ) and (φ 1 , . . . , φ m ) ∈ SL 2 (C) m , this action is defined via: These actions turn V (n) and V (λ) into representations of SL 2 (C) and (SL 2 (C)) m , respectively. These are precisely the finite dimensional irreducible representations of SL 2 (C) and (SL 2 (C)) m (see Lecture 11 of [6]), and so they are the basic building blocks of the SL 2 (C) representation theory. Actions on V (n) and V (λ) can be extended to tensor products in the usual way, and in this paper we will make use of both inner and outer tensor products. We now briefly discuss tensor product actions for those less familiar.
The outer tensor product of V (λ k ), denoted V (λ 1 ) · · · V (λ m ), is a representation of (SL 2 (C)) m with action by (φ 1 , . . . , φ m ) on simple tensors given as follows: This implies that V (λ) and V (λ 1 ) · · · V (λ m ) are isomorphic as representations, and this fact will be used when we define the symbol in Sect. 3.3.
The inner tensor product of V (λ k ), denoted V (λ 1 ) ⊗ · · · ⊗ V (λ m ), is a representation of SL 2 (C) with action by φ on simple tensors given as follows: While V (λ) and V (λ 1 )⊗· · ·⊗V (λ m ) are isomorphic as vector spaces, they are representations of different groups ((SL 2 (C)) m and SL 2 (C) respectively). The inner tensor product relates to invariants of multiple polynomials with respect to a single SL 2 (C) action. For instance, the apolarity form takes two distinct polynomials as input, and it is a classical result that this form is invariant with respect to a single action by Möbius transformation. As it turns out, this form can be viewed as an SL 2 (C)-invaiant map on an inner tensor product of polynomial spaces. It will therefore be important for us to understand these inner tensor products in a little more detail.

An important invariant map, and apolarity
To aide in our investigation of inner tensor products of SL 2 (C) representations, we now define an important SL 2 (C)-invariant linear map, denoted by D. This map has a long history in invariant theory, and we touch on this below.

Proposition 3.1 The linear map D
Proof It suffices to check this on simple tensors. Fix φ = a b c d ∈ SL 2 (C), p ∈ V (n + 1), and q ∈ V (m + 1). We compute: Powers of the D map actually appear in the literature under a few different names. The first comes from invariant theory, where the application of the map to polynomials p ∈ V (n) and q ∈ V (m) is called the r th transvectant of p and q. This map is also the result of the r th iteration of Cayley's process. These notions are discussed, for example, in chapters 4 and 5 of [13], where they are used to explicitly compute invariants and covariants of forms. In particular, the invariance of the Jacobian (1st transvectant map applied to p ⊗ q) and the Hessian (2nd transvectant map applied to p ⊗ p) can be determined in this way.
Additionally, the nth transvectant of p, q ∈ V (n) is used to define a notion of apolarity (see, e.g., [5] and [4]), and this notion corresponds to the classical one used in Grace's theorem. In fact, one of the original formulations of Grace's theorem can be found in Grace and Young's 1903 book, The Algebra of Invariants [9]. This suggests a connection between invariant theory and the analytic consequences of apolarity theory via the D map, and we will indeed see this map play a crucial role in the proof of Grace's theorem (Theorem 5.1).
We are now ready to define the homogeneous apolarity form via the D map. This form and its SL 2 (C)-invariance are then the next main step toward the definition of the symbol of an operator. In the next section, we will use this bilinear form to define an important construction called the dual of a representation. This will serve as the link to viewing spaces of linear operators as representations themselves.

Definition 3.3 We call the nth transvectant
. This is the unique (up to scalar) nondegenerate SL 2 (C)-invariant bilinear form on V (n), and therefore it is the homogenization of the classical apolarity form.
We now want to extend this definition to act on V (λ) ⊗ V (λ) for λ ∈ N m 0 . Note that for p ∈ V (λ), we have m pairs of variables given by p(x 1 , y 1 , . . . , x m , y m ), which allows us to naturally define: With this, we can define the apolarity form for V (λ) as follows.

Definition 3.4 We call the map
. This is the unique (up to scalar) nondegenerate (SL 2 (C)) minvariant bilinear form on V (λ), and therefore it is the homogenization of the multivariate apolarity form defined by Borcea and Brändén in [2].
we will often consider the maps D n and D λ to output a element of C. And as a final note, we do not justify here the claims of uniqueness and nondegeneracy stated above. Proving these claims involves decomposing V (n) ⊗ V (n) and V (λ) ⊗ V (λ) into their irreducible components, and we leave this work to Appendix A for the interested reader (see Corollaries A.9 and A.12 specifically).

The symbol of an operator
Given representations V (λ) and V (α) (for λ ∈ N m 0 and α ∈ N l 0 ) of (SL 2 (C)) m and (SL 2 (C)) l respectively, the space of linear maps between V (λ) and V (α) can be viewed as a representation of (SL 2 (C)) m+l in a standard way. This space of linear maps is denoted Hom(V (λ), V (α)). As discussed previously, we will now use the apolarity form defined above to construct a representation isomorphism between Hom(V (λ), V (α)) and V (λ α) (which is a space of polynomials in m + l variables). This will lead us to a natural definition for the symbol of an operator.
The significance of this isomorphism will come from the fact that stability results about V (λ α) will transfer to Hom(V (λ), V (α)) via the symbol lemma (Lemma 3.7) stated below. We will see in Sect. 6.2 that this lemma and Grace's theorem almost immediately imply a characterization of stability-preserving operators which is similar to that of Borcea and Brändén.
To this end, consider the standard representation isomorphism Hom We omit here the details regarding explicit definitions of the action of (products of) SL 2 (C) on Hom and dual representations. Instead, we utilize the fact that the apolarity form provides an (SL 2 (C)) m -invariant isomorphism between V (λ) and the dual representation V (λ) * , as stated in the following result.

Proposition 3.5 For any
We use the apolarity form to determine the isomorphism. In particular, up to scalar (x μ y λ−μ ) * maps to an element p ∈ V (λ) such that (x μ y λ−μ ) * = D λ ( p ⊗ ·). We compute: With this, we consider the following string of (SL 2 (C)) m+l -invariant isomorphisms: The first map is the standard isomorphism discussed above, the second map is induced by the previous proposition, and the third map is given by the discussion of outer tensor products in Sect. 3.1. This string of maps is explicitly defined on a given linear operator via: Here, T acts only on the x and y variables, and z and w are the λ variables in V (λ α). This gives the desired isomorphism between Hom(V (λ), V (α)) and V (λ α), and hence we refer to this map as the Symb map.

Definition 3.6
For λ ∈ N m 0 and α ∈ N l 0 , we define the following (SL 2 (C)) m+l -invariant isomorphism: We call Symb(T ) the (universal) symbol of T .
This expression bears striking resemblance to the symbol used by Borcea and Brändén in [1], which motivates the use of the name "symbol" here. (In fact, Symb is almost the homogenization of the Borcea-Brändén symbol.) In Sect. 6, Symb will allow us to reduce the study of Hom(V (λ), V (α)) to the study of V (λ α) via the next lemma. We refer to this next result as the symbol lemma, and it demonstrates the fundamental connection between an operator T , its symbol, and the apolarity form. Note that the computation done here in the proof of this lemma is in a sense redundant. The operator Symb was essentially defined such that Symb(T ) acts as T via D λ .
Proof Letting q μ be the coefficient of the x μ y λ−μ term of q, we compute:

Polynomial stability theory
More generally, p is said to be -stable if it doesn't vanish in . As above, we say p is weakly -stable if possibly p ≡ 0. Most all results related to zero location of polynomials then can be translated into statements about stability properties of polynomials and stability preservation properties of operations applied to polynomials.
A linear operator T is said to preserve weak -stability if T ( p) is -stable or identically zero for all -stable p. Further, a real linear operator T preserves weak real stability if the same holds for real stable polynomials. In [1], Borcea and Brändén were concerned with classifying such weak stability preserving operators. As seen in their main characterization results (Theorems 1.1 and 1.2), allowing the zero polynomial leads to a degeneracy condition in their characterization.
In order to remove this condition, we define a slightly different notion of stability: we say a linear operator T preserves (strong) -stability if T ( p) is stable and nonzero for all stable p. Similarly, we say a real linear operator T preserves (strong) real stability if the same holds for real stable polynomials. Most of the main results of this paper rely on this notion of strong stability preservation, and we will demonstrate how it relates to weak stability preservation in Sects. 6.3 and 7.3.

Polar derivatives
A crucial tool of classical stability theory is the polar derivative. In particular, this notion leads to Laguerre's theorem (Proposition 4.6), which is the main lemma toward Grace's theorem. By passing to homogeneous polynomials the polar derivative becomes conceptually simpler, and this in turn sheds further light on the general connection to SL 2 (C)-invariance and the D map. One example of this, as we will see below, is that the polar derivative can be defined as the conjugation of ∂ x by some SL 2 (C) action.
Given some "pole" x 0 ∈ C, the polar derivative with respect to x 0 of f ∈ C n [x] is classically defined as follows. ( Noticing that the term of degree n cancels out, the resulting polynomial is of degree n − 1. It is typically said that this operator generalizes the ordinary derivative in the sense that However, this operator also generalizes the ordinary derivative in more natural way, which we see by passing to V (n).
we then define the polar derivative with respect to φ as follows.
With this, the pole of φ should be interpreted as the element of CP 1 that φ sends to ∞ = (−1 : 0). This definition of the polar derivative with respect to φ is at very least a natural one, as it can be simply described as the conjugation of ∂ x by the action of φ. The following result then shows that this is actually the correct definition.
Proof Straightforward computation.
As mentioned above, d φ depends only on (−d : c), the pole of φ. So given any pole in CP 1 , we can actually choose φ ∈ SL 2 (C) to be a rotation of the Riemann sphere (i.e., CP 1 ). This then gives the following intuitive description of the polar derivative.

Projective convexity and Laguerre's theorem
Circular regions play a key role in Grace's theorem and its corollaries. The main reason for this is Laguerre's theorem, which essentially says that polar derivatives with respect to points of a circular region preserve stability for that circular region. This theorem in turn relies on the Gauss-Lucas theorem, which deals with convex regions.
A circular region in C is defined to be a disc, half-plane, or complement of a disc, and such a circular region can be either open or closed. The generalization of circular regions to CP 1 is the obvious one. A circular region in CP 1 is defined to be the sets in CP 1 for which the stereographic projection is a circular region in C. Note that SL 2 (C) acts transitively on the set of all circular regions in C or in CP 1 . We now state a lemma to Laguerre's theorem, which gets at the heart of the importance of circular regions.

Lemma 4.3
Let C ⊆ CP 1 be a circular region, and let φ ∈ SL 2 (C) be such that its pole is not in C. Then, the stereographic projection of φ · C is convex.
Then, φ maps (x 0 : y 0 ) to ∞ ∈ CP 1 and maps C to another circular region. Since (x 0 : y 0 ) / ∈ C implies ∞ / ∈ φ · C, the sterographic projection of φ · C is either an open half-plane or is bounded away from ∞. Since φ · C is a circular region, it must be convex.
This then leads to a natural extension of the notion of a circular region.

Definition 4.4 Given
We now classify all projectively convex sets in CP 1 in the following. This result has been demonstrated before in [18], where projectively convex regions are referred to as generalized circular regions. So, one example of a projectively convex set which is not quite a circular region is H + ∪R + . Another is H + ∪ [0, 1]. Yet another (albeit after a bit of consideration) is . We now state a homogeneous version of Laguerre's theorem, extended to projectively convex sets. Proposition 4.6 (Laguerre) Let C ⊆ CP 1 be projectively convex, and fix φ ∈ SL 2 (C). If the pole of φ is in C, then d φ preserves strong C-stability.
Proof Gauss-Lucas and the fact that CP 1 \C is projectively convex give the result. Specifically, for C-stable p ∈ V (n) consider φ · p, which is stable in φ · C ∞. Letting B be the complement of C, the dehomogenization of this polynomial is then of degree exactly n with all of its roots in the stereographic projection of φ · B. By projective convexity, φ · B is convex and therefore Gauss-Lucas implies ∂ x (φ · p) is φ · C-stable and not identically zero.
Corollary 4.7 Let C k ⊆ CP 1 be projectively convex regions for k ∈ [m], and fix φ ∈ SL 2 (C). If the pole of φ is in C k 0 , then d φ acting on the variables (x k 0 , y k 0 ) preserves strong (C 1 × · · · × C m )-stability.
Proof Follows from the fact that taking derivatives in some variables commutes with evaluation in the others. Specifically, p ∈ V (λ) is (C 1 × · · · × C m )-stable iff p = 0 for all evaluations in C 1 × · · · × C m . So, evaluating p in all variables in that product of sets except (x k 0 , y k 0 ) gives us a C k 0 -stable polynomial in V (λ k 0 ). Applying the previous proposition then gives the result.

Real stable polynomials
We now give a number of classical real stability results, along with a few results from [1] and [2]. Additionally, we state these results for homogeneous polynomials in V R (λ), taking roots at infinity into account. The results of this section will come in to play mainly in Sect. 7, where we discuss real linear operators and operators preserving interval-and ray-rootedness.
The first result we will need for our considerations of V R (λ) is a version of the Hermite-Biehler theorem, often called the Hermite-Kakeya-Obreschkoff theorem. We state here without proof the multivariate version essentially used in Theorem 1.9 of [1] (see also §2.4 of [16]). First we need a definition.
This result will be crucial to our consideration of real polynomials and real stability (as it was in [1]). Its main use for us in this direction is made explicit in the following.
Proof By the Hermite-Biehler theorem, there exist q, r ∈ V R (λ) such that p = q + ir and aq + br is real stable or zero for all a, b ∈ R. So, aT (q) + bT (r ) is real stable or zero for all a, b ∈ R. By Hermite-Biehler again, The next two results are from [1], the first of which gives an equivalent characterization for a polynomial to be a scalar multiple of a real stable polynomial. This result will be specifically used in Sect. 7 to generalize complex operator theoretic stability results to the real stability case. The next result provides the degeneracy cases in the Borcea-Brändén characterizations (recall the dimension restrictions of Theorems 1.1 and 1.2). We will use this result to explicate the link between our operator characterization and the Borcea-Brändén characterization (see Lemmas 6.11 and 7.3).

Lemma 4.12 [1, Lemma 3.2]
Let W ⊆ V K (λ) be a K-vector subspace (for K = C or K = R) consisting only of weakly stable (resp. weakly real stable) polynomials. We have: By applying appropriate Möbius transformations, note that (a) of the above lemma can be generalized to (C 1 × · · · × C m )-stable polynomials for any open circular regions C 1 , . . . , C m ⊆ CP 1 .
We now state the last result of this section, which refines the Hermite-Biehler theorem for top-degree monic polynomials in V R (n). This refinement comes through the notion of interlacing polynomials and is much closer to the original statement of the classical Hermite-Biehler theorem (e.g., see Theorem 6.3.4 in [15]).

Lemma 4.13
For top-degree monic p, q ∈ V R (n), p q if and only if the roots of p and q (denoted in increasing order by (α k : 1) and (β k : 1), respectively) interlace on the real line in the following way: Further, if these equivalent conditions hold, then gives a total order on the top-degree monic elements of the span of p and q in V R (n). This order is equivalently defined via the order of the kth largest roots, for any k ∈ [n] such that α k = β k .
Proof The fact that p q is equivalent to interlacing roots is the classical univariate Hermite-Biehler theorem. That q has larger roots than p can be obtained by the fact that the (n − 1)st derivative of q + i p must be H + -stable. Since both polynomials are top-degree monic, this (n − 1)st derivative will be a complex linear combination of two linear terms. This complex linear combination is given as follows, where s q and s p denote the respective sums of the roots of q and p: Since this polynomial is H + -stable, it must be that s q +is p n(1+i) ∈ H − . We further compute: Therefore s q ≥ s p , which is the same as saying that the sum of the roots of q is larger than that of p. Since we already know that the roots of q and p interlace, this implies that q has larger roots than p.
As for the total ordering property, let r and s be two polynomials in the real span of p and q. Any real linear combination of these polynomials is then a real linear combination of p and q (and hence is real-rooted), and Hermite-Biehler implies either r s or s r . By the above interlacing condition, it is straightforward to see that this total order is given by looking at the order of the kth roots, for any k ∈ [n].
immediate once Grace's theorem has been proven, and yet will quickly yield stronger results regarding linear operators in the next section.
In the usual proof of the classical univariate Grace's theorem, reference to linear factors of f ∈ C n [x] is necessary. This makes generalization to C λ [x 1 , . . . , x m ] difficult, as multivariate polynomials do not necessarily have any linear factors. In our new proof, we are able avoid reference to linear terms by using particular features of the D map. This means that our proof method works for any λ.
Theorem 5.1 Fix λ ∈ N m 0 and p, q ∈ V (λ). Also, denote C := H + ∪R + and C := H − ∪R − , where the closures are considered to be in CP 1 . If p is C m -stable and q is C m -stable, then D λ ( p ⊗ q) = 0.
Proof We prove the theorem by induction on degree. For λ ≡ 0, the result is obvious. For |λ| ≥ 1, we can assume WLOG that λ 1 ≥ 1 by permuting the variables. Define δ 1 : =  (1, 0, 0, . . . , 0) ∈ N m 0 . Since C and C are projectively convex, Corollary 4.7 implies (a∂ By induction and the stability properties discussed above, we have However, we can pick α ∈ H + ∪ R + such that α 2 is any value of C\{0} we want, including that of . This contradiction gives the result.

Other regions
We now generalize the above theorem to other regions via SL 2 (C) action and topological considerations. Theorem 5.7 can then be considered our most general form of Grace's theorem. First though, we define two new notions in order to simplify the rest of this section.

Definition 5.2
Fix m ∈ N 0 and any sets S 1 , S 2 ⊆ (CP 1 ) m . We call (S 1 , S 2 ) a Grace pair if: for all λ ∈ N m 0 and p, q ∈ V (λ) such that p is S 1 -stable and q is S 2 -stable, we have that D λ ( p ⊗ q) = 0. That is, if Grace's theorem holds for S 1 and S 2 . Definition 5. 3 We say that a Grace pair is disjoint if it is of the form (C 1 × · · · × C m , B 1 × · · · × B m ) and C k and B k are disjoint for all k ∈ [m].
This yields the following restatement of the above theorem.

Corollary 5.4 For any m
The sets considered above intersect at 2 points (0 and ∞), and this ends up being crucial to the proof. So, in order to extend to the full generality of Grace's theorem, we will need to find such points even when the stability sets of two polynomials p and q do not a priori intersect at all. To this end, we give the following lemmas.

Lemma 5.5
Fix λ ∈ N m 0 and any closed circular regions C 1 , . . . , There exist open circular regions U 1 , . . . , U m such that C k ⊂ U k for all k ∈ [m] and p is (U 1 × · · · × U m )-stable.
Proof Follows from compactness of CP 1 and closedness of C 1 × . . . × C m and of the zero set of p.
For the next lemma, note that the boundary of any circular region C is topologically equivalent to the unit circle in C (i.e., the boundary of the unit disc). With this, we call a Proof Let ∂C • denote the boundary of (the closure of) C • , and let S ⊆ CP 1 be the intersection of ∂C • and the zero set of p. Since the zero set of p is closed, we have that S is closed in ∂C • . And further, γ ∩ S = ∅ by assumption. Defining to be the connected component of ∂C • \S containing γ then gives the result.
Using these lemmas and the SL 2 (C)-invariance of the apolarity form, we obtain the following generalization of Grace's theorem. Here, (ii) and (iii) give the multivariate Grace's theorem proven in [2].

Theorem 5.7
For m ∈ N 0 and C 1 , . . . , C m , B 1 , . . . , B m ⊆ CP 1 , we have that (C 1 × · · · × C m , B 1 × · · · × B m ) is a Grace pair for the following regions. Proof (i). By Proposition 4.5, every projectively convex region in CP 1 is the union of an open circular region and a portion of its boundary. Since C k ∪ B k = CP 1 and C k ∩ B k is exactly two points, we then must have that C k = φ k · (H + ∪ R + ) and B k = φ k · (H − ∪ R − ) for some φ k ∈ SL 2 (C). Since D λ is (SL 2 (C)) m -invariant, the result follows from Theorem 5.1.
(ii). Fix p, q ∈ V (λ). If p is (C 1 × · · · × C m )-stable and q is (B 1 × · · · × B m )-stable, then Lemma 5.5 implies p is (U 1 × · · · × U m )-stable for some open circular regions U 1 , . . . , U m such that C k ⊂ U k for all k ∈ [m]. Since C k ∪ B k = CP 1 , we then have that U k ∩ B k is open and nonempty. Since U k and B k are circular regions, their intersection in fact contains an open annulus or open strip in C. Therefore we may slightly shrink U k and B k to get closed circular regions U k and B k such that U k ∪ B k = CP 1 and U k ∩ B k = ∂U k = ∂ B k , where ∂ B k denotes the boundary of B k . We can then further remove portions of the respective boundaries of U k and B k to get projectively convex regions U k and B k such that U k ∪ B k = CP 1 and U k ∩ B k is exactly two points. Since U k ⊂ U k and B k ⊂ B k , we have that p is (U 1 ×· · ·×U m )-stable and q is (B 1 × · · · × B m )-stable. Therefore (i) implies D λ ( p ⊗ q) = 0, and this implies (ii).
(iv). Let p, q ∈ V (n) be such that p is C 1 -stable and q is B 1 -stable. Defining B 1 := CP 1 \C 1 ⊆ B 1 , we further have that q is B 1 -stable. So WLOG we may assume that B 1 = B 1 . Note that this implies ∂C • 1 = ∂ B • 1 ; that is, the boundaries coincide. If C 1 is a circular region then so is B 1 , and therefore D n ( p ⊗ q) = 0 by (ii) or (iii). This implies (iv) in this case.
Otherwise by Proposition 4.5, we have that we then further can find closed subsets 1 ⊂ 1 and 1 ⊂ 1 such that 1 ∪ 1 = ∂C • 1 and ∩ is exactly two points. Therefore D n ( p ⊗ q) = 0 by (i), and this implies (iv).

Notice that (ii) and (iii) in this result do not allow for mixed open and closed stability regions.
That is, all of the C k must be open and all of the B k closed, or vice versa. We show that this particular point cannot be ignored, using the following example. λ = (1, 1, 1), denote E := CP 1 \D, and consider the polynomial p := x 1 x 2 x 3 − y 1 y 2 y 3 = Hmg λ (x 1 x 2 x 3 − 1). First, it is easy to see that D λ ( p ⊗ p) = 0. Also, p is D 3 -stable and E 3 -stable, but it is not D 3 -stable nor As for whether or not the two-point intersection condition can be removed from (i) seems to be a more subtle point. It would be quite nice if this condition could be removed, but it is unclear whether or not it is possible.

Evaluation symbols
One way to interpret the stability properties of a given polynomial is via the stabilitypreservation properties of a particular type of linear operator: the evaluation map. That is, the map which evaluates a polynomial p(x, y) at  (a 1 , b 1 , . . . , a m , b m ) ∈ C 2m (with (a j , b j ) = 0 for all j), we can define the corresponding evaluation map as an element of Hom(V (λ), V (0)) since V (0) ∼ = C. This allows us to obtain symbols for evaluation maps, and these play an important role in our linear operator characterization.  p(a, b) = p(a 1 , b 1 , . . . , a m , b m ). We call Symb(ev (a,b) ) ∈ V (λ) the evaluation symbol with root (a, b). Further: The main significance of this notion comes from the following result, which is essentially just a restatement of the symbol lemma (Lemma 3.7) for evaluation symbols. and (a, b) =  (a 1 , b 1 , . . . , a m evaluation symbol with root (a, b), we have: In what follows, we will extend Grace's theorem in a number of ways, mainly relying on the previous lemma and the symbol lemma itself. As we will see, the representation theoretic mentality combined with repeated use of the symbol lemma will yield many of the results of this paper with surprising simplicity.
We now obtain an interesting corollary of Grace's theorem, making use of the notion of a disjoint Grace pair. This particular formulation of the theorem will serve as a model for our linear operator characterization in Sect. 6.2.
(iii) ⇒ (i) This follows immediately from the definition of Grace pair (Definition 5.2).

Stability properties of complex linear operators
In [1], Borcea and Brändén were concerned with classifying the class of weak -stability preserving operators, where is some product of open circular regions. What they found is that an operator preserves weak -stability if a particular associated polynomial (what they called the symbol) is -stable. However, the "only if" direction does not necessarily hold. In particular, there are some weak -stability preserving operators for which the corresponding symbol is not -stable. They then showed that this could only happen under very specific circumstances: the operator must have image of dimension at most one.
Here, we will characterize all strong -stability preserving linear operators (for a bit more general ), as well as linear operators which map between different stability regions. And, as it turns out, the extra premise of strong stability preservation is exactly what is needed to have symbol stability be an equivalent condition. In a way, this makes sense: weak -stability preservation counts the zero polynomial as -stable, which in turn corresponds to potential zeros of the symbol in the region of stability. This does not happen with strong stability preservation, allowing for a more straightforward characterization.
First though, let's take a closer look at the Borcea-Brändén characterization of weak stability-preserving linear operators.

Weak stability preservation
Borcea and Brändén define the following symbol: They then obtain the following characterization of stability-preserving linear operators. (i) T maps H m + -stable polynomials to weakly H m + -stable polynomials. (ii) One of the following holds:

has image of dimension at most one, and is of the form
where q ∈ C[x 1 , . . . , x m ] is H m + -stable, and ψ is some linear functional.
Using our terminology, this is a characterization of weak stability-preserving linear operators. This fantastic result perhaps has but one unfortunate piece: the degeneracy condition (ii)(b). Its necessity is demonstrated in the following. a k x k → (a n + a n−2 )x n This operator obviously preserves weak H + -stability. We then have that Symb B B (T ) = (z 2 + 1)x n , which is not H 2 + -stable.
As we will see below, this condition can be removed once we only consider strong stabilitypreserving operators. So then, maybe strong stability is the more natural notion? However "natural" it is, unfortunately it leaves out operators one might wish to consider. The most fundamental of such operators is the derivative operator ∂ x . While ∂ x preserves strong H +stability, it only preserves weak H + -stability. Specifically, 1 ∈ C n [x] is H + -stable (all its roots are at ∞), but ∂ x 1 ≡ 0. With this, one obviously wants to be able to include weak stability preserving operators in any characterization of H + -stability preserving operators. We discuss how to use our strong stability preservation characterization to deal with operators like ∂ x in Example 6.8.
One should notice the generality of this result in terms of stability regions. First note that any disjoint Grace pair can be considered, without altering the symbol in any way (e.g., via conjugation by Möbius transformations). And further, the output sets that can be considered have no restrictions whatsoever. The power of these extra features can be seen in the following examples, which demonstrate classical results regarding polynomial convolutions in a very symbol-oriented way.

Example 6.3
Fix p, q ∈ V (n), so that (z j : 1) are the roots in CP 1 of q for j ∈ [n]. So, q has no roots at ∞. The additive (Walsh) convolution of p and q is defined via: With this, T q ( p) := p * n + q is a linear operator in Hom(V (n), V (n)), and we have: ⎤ ⎦ Let C ⊂ CP 1 be any projectively convex region, and define S := j (C + z j ). If we order the input variables of Symb(T q ) as (z, w, x, y), it is then straightforward to show that Symb(T q ) is C × (CP 1 \S)-stable. (First deal with possible (x : y) = (1 : 0) or (z : w) = (1 : 0) cases, and then assume y = w = 1 to simplify the remaining cases.) Applying the previous theorem, this implies T q maps polynomials with roots in C to polynomials with roots in S. (This is Theorem 5.3.1 in [15].) Picking C = H − and real-rooted q implies T q maps H + -stable polynomials to H + -stable polynomials. Restricting to p ∈ V R (n) then shows that T q preserves real-rootedness.

Example 6.4
Fix p, q ∈ V (n), so that (z j : 1) = 0 are the roots of q for j ∈ [n]. So, q has no roots at 0 or ∞. The multiplicative (Grace-Szegő) convolution of p and q (with coefficients p k and q k , respectively) is defined via: With this, T q ( p) := p * n × q is a linear operator in Hom(V (n), V (n)), and we have: Let C ⊂ CP 1 be any projectively convex region, and define S := j (z j · C). If we order the input variables of Symb(T q ) as (z, w, x, y), it is then straightforward to show that Symb(T q ) is C × (CP 1 \S)-stable. (As above, first deal with possible (x : y) = (1 : 0) or (z : w) = (1 : 0) cases, and then assume y = w = 1 to simplify the remaining cases.) Applying the previous theorem, this implies T q maps polynomials with roots in C to polynomials with roots in S. (This is Theorem 3.4.1d in [15].) Picking C = H − ∪R + and q with only positive roots implies T q maps (H + ∪ R − )-stable polynomials to (H + ∪ R − )-stable polynomials. Restricting to p ∈ V R (n) then shows that T q preserves positive-rootedness.
In order to prove the above theorem, we need an operator-theoretic corollary to Grace's theorem. The following result is the main motivation for the symbol lemma (Lemma 3.7), and demonstrates just how closely Grace's theorem relates to stability properties of linear operators. Further, it gives a slightly stronger result in one direction of the above characterization, as Grace pair disjointness is not a required premise. V (α)), any Grace pair (C 1 × · · · × C m , B 1 × · · · × B m ), and any sets S 1 , . . . , The evaluation symbol lemma (Lemma 5.10) and the symbol lemma (Lemma 3.7) then give us the following expression of T (q) evaluated at (c, d) = (c 1 , d 1 , . . . , c l , d l ): In the last expression above, D λ acts on the variables (z, w) = (z 1 , w 1 , . . . , z m , w m ). Since r (z, w) := Symb (T )(z, w, c, d) is (B 1 × · · · × B m )-stable and q(z, w) is (C 1 × · · · × C m )stable, we have that the last expression above is nonzero by definition of Grace pair (Definition 5.2). This implies T (q) is (S 1 × · · · × S l )-stable.
With this, we now give the proof of Theorem 6.2.

Proof of Theorem 6.2
The statement of this result, as well as its proof, is quite similar to that of the evaluation symbol version of Grace's theorem given in Corollary 5.11. We explicitly give the proof anyway, as it is rather short and straightforward.
As mentioned above, the previous proposition gives a slightly stronger result in the (symbol stability ⇒ operator stability) direction. Using it, we revisit the additive and multiplicative convolutions with a more algebraic/symbolic mentality. Example 6.6 By Definition 3.6, the Symb map gives a bijection between certain spaces of linear operators and polynomials. So, we can uniquely define a linear operator by giving its symbol. Using this idea, we specify T ∈ Hom (V (n, n), V (n)) by defining its symbol in V (n, n, n) with variables (z, w), (t, s), (x, y) as follows: Symb(T ) := Hmg (n,n,n) (x − (z + t)) n = (xws − (zs + tw)y) n Now, let us consider the additive convolution * n + as an element of Hom(V (n, n), V (n)) in the following way. Since V (n, n) ∼ = V (n) V (n), we define * n + on elements p q ∈ V (n) V (n) via * n + ( p q) := p * n + q and extend linearly. We then compute Symb( * n + ) as follows: That is, * n + is the operator that has our desired symbol. Fixing any a, b, c, d ∈ R such that a < b and c < d, we define the sets all their roots in (a, b) and (c, d), respectively. For real-rooted p, q of degree n, this implies: Notice that we actually get a bit more. For (C 1 ×C 2 )-stable r := j p j q j ∈ V (n) V (n) ∼ = V (n, n), we have that * n + [r ] is S-stable. That is, * n + has stability properties as an operator in Hom(V (n, n), V (n)), not just as a convolution between two polynomials in V (n). Example 6.7 As in the previous example, we can consider the multiplicative convolution * n × as an element of Hom(V (n, n), V (n)) by defining * n × ( p q) := p * n × q on elements p q ∈ V (n) V (n) ∼ = V (n, n) and extending linearly. We then compute its symbol in V (n, n, n) with variables (z, w), (t, s), (x, y) as follows: Fixing any a, b, c, d ∈ R + such that 0 < a < b and 0 < c < d, we define the sets C 1 , C 2 , B 1 , and B 2 as in the previous example. We then define S := R\[ac, bd]. Proposition 6.5 then implies p * n × q has all its real roots in [ac, bd] whenever p, q ∈ V R (n) have all their roots in (a, b) and (c, d), respectively. (Notice that we could not apply the proposition if H + ⊂ S or H − ⊂ S.) Since Example 6.4 implies p * n × q is positive-rooted (and hence, real-rooted) whenever p and q are, this implies: As in the previous example, we also obtain stability properties for * n × as an operator in Hom(V (n, n), V (n)), and not just as a polynomial convolution.
Using similar techniques, we can also circumvent the issue that arises from the fact that ∂ x only preserves weak stability. Example 6.8 For fixed n ≥ 1, consider the operator ∂ x ∈ Hom(V (n), V (n−1)). We compute: ∪ (a, b) and B := H + \ (a, b), where the variables are ordered (z, w), (x, y). (Notice that this does not hold when ∞ ∈ C, due to the w factor in the symbol.) Since (C, B) is a disjoint Grace pair, the Theorem 6.2 implies ∂ x preserves strong B-stability.
With this, let f ∈ C n [x] be a H + -stable polynomial of degree 1 ≤ m ≤ n, and let p ∈ V (m) be its degree-m homogenization. Then p has no roots at infinity, and therefore there exists a < b such that p is H + \(a, b) -stable. The previous discussion implies ∂ x p is H + \(a, b) -stable, and in particular ∂ x p is H + -stable. Since ∂ x commutes with homogenization, this also implies ∂ x f is H + -stable.
Other issues related to weak stability preservation can be dealt with in a similar way, by considering stability regions with small intervals in R about ∞ attached. More generally though, the Borcea-Brändén characterization ends up being a corollary of Theorem 6.2, which we discuss and demonstrate now.

Deriving the complex Borcea-Brändén characterization
As mentioned above, we hope to obtain the Borcea-Brändén characterization from our strong stability characterization given in Theorem 6.2. To this end, we state two corollaries to Theorem 6.2, which look (naively) as close to the Borcea-Brändén characterization as possible. Let C c denote the complement of C in CP 1 . Corollary 6.9 Fix λ, α ∈ N m 0 , a linear operator T ∈ Hom(V (λ), V (α)), and a Grace pair of the form (C 1 × · · · × C m , C c 1 × · · · × C c m ). The following are equivalent. (i) T preserves strong (C 1 × · · · × C m )-stability.
(ii) Symb(T ) is (C c 1 × · · · × C c m ) × (C 1 × · · · × C m )-stable. Corollary 6.10 Fix λ, α ∈ N m 0 and a linear operator T ∈ Hom(V (λ), V (α)). T preserves strong stability iff Symb(T ) is (H − m × H m + )-stable. In Theorem 1.1, the analogous "if" direction of the previous corollary is paraphrased as follows: T preserves weak stability if the Borcea-Brändén symbol of T is stable. To see how this statement relates, we restate the definition of the Borcea-Brändén symbol: Notice that by applying z → −z and homogenizing, we obtain (up to scalar) the universal symbol Symb(T ) defined in this paper. The crucial difference then is the fact that the Borcea-Brändén "if" direction deals only with open upper half-planes, whereas the previous corollary requires closed half-plane stability of Symb(T ) in the first m pairs of variables. That is, the required premises of the "if" direction of the previous corollary are strictly stronger than that of the Borcea-Brändén result. These two results can be reconciled, however, which we now demonstrate. The following result provides the main link to the Borcea-Brändén characterization, and it can be intuitively described as follows: with the exception of having a one-dimensional range, a linear operator which maps (C 1 ×· · ·×C m )-stable polynomials to weak (B 1 ×· · ·× B m )-stable polynomials can only have zeros on the boundary of the set of (C 1 × · · · × C m )-stable polynomials. (i) T maps (C 1 ×· · ·×C m )-stable polynomials to weakly (B 1 ×· · ·×B m )-stable polynomials.
Proof By appropriate SL 2 (C) action, we can assume WLOG that C k = B k = D, the unit disc, for all k ∈ [m]. This lemma then yields the following corollaries to Theorem 6.2. Applying the necessary maps to convert Symb to Symb B B as discussed above, these results give precisely the Borcea-Brändén characterization proven in Theorem 1.1 and more generally in Theorem 6.3 of [1]. In particular, Corollary 6.12 can be seen as a unification of the complex characterization results of [1]. (i) T preserves weak (C 1 × · · · × C m )-stability.
Proof The result follows from the Lemma 6.11 and Theorem 6.2 applied to an operator T which maps (C 1 × · · · × C m )-stable polynomials to nonzero (C 1 × · · · × C m )-stable polynomials.
for some weakly stable polynomial p 0 ∈ V (α) and some linear functional ψ.
Notice that our naive guess at strong stability results which emulate the Borcea-Brändén characterization (Corollaries 6.9 and 6.10) was incorrect. We actually needed to consider closed circular stability regions C k , so that their complements in CP 1 would be open (i.e., to ensure Grace pair disjointness, which is required to apply Theorem 6.2). We see this play out in condition (ii)(a) of Corollary 6.12.

Stability properties of real linear operators
Borcea and Brändén also classified the class of weak real stability preserving linear operators. As in the complex case, they showed that weak real stability preservation of a linear operator T is almost equivalent to real stability of the associated symbol Symb B B (T ). We have to say "almost equivalent" here because there are certain weak real stability preserving operators for which the corresponding symbol is not real stable. As before, this implies a certain dimension restriction: such operators must have image of dimension at most two.
We will now characterize all strong real stability preserving linear operators. As above, strong real stability preservation will serve to eliminate the degeneracy condition of the Borcea-Brändén characterization. In this section, we duplicate the outline of our previous discussion on complex operators, making use of arguments similar to those found in [1] to fill in the gaps.
Further, we also obtain a characterization of a certain class of operators which preserve ray-and interval-rootedness. The question of a full characterization of such operators is as of yet still an open problem (see [3]). Here, we answer this question for operators which preserve both strong ray-or interval-rootedness as well as weak real-rootedness.

Weak real stability preservation
Borcea and Brändén obtain the following characterization of weak real stability preserving linear operators. Recall the notion of proper position (denoted by ) given in Definition 4.8. (i) T maps real stable polynomials to weakly real stable polynomials.
(ii) One of the following holds:

has image of dimension at most two, and is of the form
where q, r ∈ R[x 1 , . . . , x m ] are weakly real stable such that q r , and ψ 1 , ψ 2 are real linear functionals.
As in the case of complex operators, the degeneracy condition (ii)(c) is the result of allowing weak real stability preserving operators. We now give an example which demonstrates its necessity.

Deriving the real Borcea-Brändén characterization
As in the complex case, we now obtain the Borcea-Brändén weak real stability characterization as a corollary to our strong real stability characterization given in Theorem 7.2. To this end, we start by giving a sort of real stability version of Lemma 6.11. The proof of this lemma is similar in spirit to that of the strong real stability characterization given above.
where q, r ∈ V R (α) are weakly real stable such that q r , and ψ 1 , ψ 2 are real linear functionals.
Proof (i) ⇒ (ii). By the complex characterization (Theorem 6.2), we only need to consider evaluation symbols when demonstrating (a) or (b). For any (z 0 : (ii) ⇒ (i). If (c) holds, then (i) follows from Hermite-Biehler. Otherwise, suppose WLOG that (a) holds. We can then use an argument similar in spirit to that of Lemma 6.11 to show that T maps H m + -stable polynomials to weakly H m + -stable polynomials. Since T restricts to a real operator, this implies (i).
As in Lemma 6.11, we use the previous lemma to link the characterizations of weak and strong stability preserving operators as follows. Applying the necessary maps to convert Symb(T ) to Symb B B (T ) below gives essentially the characterization of weak real stability preserving operators given in Theorem 1.2.

Corollary 7.4
Fix λ, α ∈ N m 0 and a linear operator T ∈ Hom(V (λ), V (α)) such that T restricts to a real linear operator from V R (λ) to V R (α). The following are equivalent.
(i) T preserves weak real stability.
(ii) One of the following holds:

c) T has image of dimension at most two, and is of the form
where q, r ∈ V R (α) are weakly real stable such that q r , and ψ 1 , ψ 2 are real linear functionals.
Proof Apply the complex characterization (Theorem 6.2) to conditions (ii)(a) and (ii)(b) of Lemma 7.3 above.

Ray and interval stability
We now apply the above results to projectively convex regions of the form H + ∪ J c , where J ⊂ R is some connected set. From this, we obtain a classification of operators which both preserve strong J -rootedness and weak real-rootedness (a polynomial p ∈ V (n) is J -rooted if all its roots lie in J ). This of course does not completely solve the open problem of providing a classification of interval-and ray-stability preserving operators (see, e.g., [3]). However, it does seem to be the natural corollary obtained by applying proof methods similar to that of [1].
That said, we now proceed to prove the main result of this subsection, Theorem 7.8. We first start with a short-hand definition in order to simplify the proof. Definition 7.5 Fix λ, α ∈ N m 0 and a linear operator T ∈ Hom(V (λ), V (α)) such that T restricts to a real linear operator and preserves weak real-stability. We say T is degenerate if it satisfies condition (ii)(c) of Corollary 7.4.
We now prove two lemmas. The first is straightforward, but rather interesting in its own right. Lemma 7.6 Fix a closed bounded interval J ⊂ R and a subspace W ⊆ V R (n) consisting of weakly real-rooted polynomials. Let S ⊆ W denote the subset of top-degree monic J -rooted polynomials. There exist p, q ∈ S such that p q and S is the convex hull of p and q.
Proof Lemma 4.12 implies W is of dimension at most two, and so then Lemma 4.13 implies the relation is a total order on S. Applying the root ordering property of Lemma 4.13, the closedness of S implies there are p, q ∈ S such that p q and p r q for all r ∈ S. Basic sign arguments and the fact that S is contained in the span of { p, q} then imply S is the the convex hull of { p, q}.
The second lemma is perhaps less straightforward in terms of proof, but follows from the following intuitive idea: an open ball in some complex subspace of polynomials yields, roughly speaking, an open ball of zeros.
Lemma 7.7 Fix n, m ∈ N 0 and a linear operator T ∈ Hom(V (n), V (m)) which restricts to a real linear operator and preserves weak real-rootedness. If there exist some H + -stable p 0 ∈ V (n) and some (x 0 : y 0 ) ∈ R such that T ( p 0 )(x 0 , y 0 ) = 0, then one of the following holds: Proof Let q 0 , r 0 ∈ V R (m) be such that T ( p 0 ) = q 0 + ir 0 . Also suppose that T ( p 0 ) ≡ 0 and that (b) does not hold, and let p 1 be such that T ( p 1 )(x 0 , y 0 ) = 0. WLOG, we may also assume p 1 ∈ V R (n) by considering its real or imaginary part. We will now prove that T ( p 0 ) must be real-rooted.
First, suppose further that T ( p 0 ) has a multiple root at (x 0 , y 0 ). For small fixed , p 0 + p 1 is H + -stable and so Lemma 4.10 implies T ( p 0 + p 1 ) is either H + -stable or H − -stable. Hermite-Biehler (Proposition 4.9) then implies q 0 + T ( p 1 ) and r 0 have interlacing roots. However, since T restricts to a real linear operator, it must be that q 0 and r 0 both have a multiple root at (x 0 , y 0 ). The fact that q 0 + T ( p 1 ) has no root at (x 0 , y 0 ) yields a contradiction, as interlacing is then impossible.
Otherwise, T ( p 0 ) has a simple root at We now prove our main result on ray-and interval-stability preserving operators. First we state the theorem for closed bounded output intervals, as it clarifies the proof quite a bit. We will then extend the result to other connected regions in R.
Theorem 7.8 Fix n, m ∈ N 0 and a linear operator T ∈ Hom(V (n), V (m)) which restricts to a real linear operator. Further, let I ⊆ R be any interval, and let J ⊂ R be any closed bounded interval. The following are equivalent.
(i) T preserves weak real-rootedness and maps I -rooted polynomials to nonzero J -rooted polynomials. (ii) One of the following holds: (c) T has image of dimension at most two, and is of the form where q, r ∈ V R (m) are top-degree monic and weakly J -rooted such that q r, and ψ 1 and ψ 2 are real linear functionals such that ψ 1 ( p) · ψ 2 ( p) ≥ 0 (not both zero) holds for any I -rooted p.
Proof (i) ⇒ (ii). Suppose T is nondegenerate. So, Symb(T ) is either (H − × H + )-stable or (H − × H − )-stable by Corollary 7.4. By Lemma 7.3, either T maps H + -stable evaluation symbols entirely to nonzero H + -stable polynomials or entirely to nonzero H − -stable polynomials. If for some (z 0 : w 0 ) ∈ H − we have that T [(w 0 x − z 0 y) n ] has a root in R, then we can apply the previous lemma. If condition (a) of the lemma holds, then T [(w 0 x − z 0 y) n ] is real-rooted or identically zero. The proof of Lemma 7.3 then implies T is degenerate, a contradiction. Otherwise condition (b) of the lemma holds, and therefore the real roots of T [(w 0 x − z 0 y) n ] must be in J . So in fact, T maps H + -stable evaluation symbols entirely to nonzero (H + \J )-stable polynomials or entirely to nonzero (H − \J )-stable polynomials. Finally, T maps I -rooted evaluation symbols to nonzero (H + \J )-stable and (H − \J )-stable polynomials by assumption. The complex characterization (Theorem 6.2) then implies (a) or (b).
Otherwise, T is degenerate and T [V R (n)] consists entirely of real-rooted polynomials. Condition (c) follows from Lemma 7.6.
(ii) ⇒ (i). By Corollary 7.4, T preserves weak real-rootedness. If (a) or (b) holds, then the complex characterization (Theorem 6.2) and the fact that T restricts to a real operator imply T maps I -rooted polynomials to nonzero J -rooted polynomials.
Otherwise (c) holds. For any real-rooted p, let λ( p) and μ( p) denote the largest and smallest roots of p, respectively. Since q, r are top-degree monic, every convex combination of q and r has all its roots in the interval [μ(q), λ(r )] ⊆ J . Since ψ 1 · ψ 2 ≥ 0 (not both zero) holds for I -rooted polynomials, we have that T maps I -rooted polynomials to nonzero J -rooted polynomials.
Notice that this result immediately holds for other closed, connected regions I , J ⊂ R by the action of some appropriate φ ∈ SL 2 (R). In fact, one can directly apply the action of φ to conditions (ii)(a) and (ii)(b), due to the fact that our definition of the "universal" symbol works for any projectively convex regions. The only significant change comes when applying φ to condition (ii)(c). Further, the only issue with (ii)(c) as it is written now is the requirement that p 1 and p 2 be top-degree monic polynomials. Having zeros at infinity, for instance, means that a polynomial cannot ever be top-degree monic (as the leading homogeneous coefficient is 0). There are ways to rewrite (ii)(c) that avoids this problem, but it is probably more intuitive to state the result as above and apply φ ∈ SL 2 (R).

Proof
The condition that the largest root of p 1 and the smallest root of p 2 are in J (and the fact that p 1 p 2 ) implies that α p 1 + β p 2 is J -rooted for all α, β > 0. Applying Lemma 7.6 to J completes the proof.
We now give a few examples. The first demonstrates the necessity of the premise that T preserves weak real-rootedness. T n : x k y n−k → Hmg n [x(x − 1)(x − 2) · · · (x − k + 1)] By Proposition 7.31 in [7], T n preserves positive-rootedness for all n. However, T 2 does not preserve real-rootedness, for example. In particular: If f has zeros at 0, then f (∂ x ) may map some nonzero [b, c]-rooted polynomials to 0.
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A Tensor product decomposition of SL 2 (C) representations
In this appendix, we discuss in detail the decomposition of inner tensor products of SL 2 (C) and (SL 2 (C)) m representations. The results given here are for the most part standard, and they are typically presented via the theory of Lie groups and algebras (e.g., in [6] and [10]). Here though, we discuss these results in terms of the polynomial spaces V (n) and V (λ).
That said, the first results we state demonstrate the importance of V (n) and V (λ) in the representation theory of SL 2 (C). In fact, these representations are precisely the irreducible representations of SL 2 (C) and (SL 2 (C)) m , respectively (see Lecture 11 of [6], and also Proposition 2.3.23 of [11]). We will not make full use of this fact but will need the following simpler results.
Proposition A.1 For all n ∈ N 0 , we have that V (n) is an irreducible representation of SL 2 (C) of dimension n + 1.
is an irreducible representation of (SL 2 (C)) m of dimension i (λ i + 1).
In particular, outer tensor products of irreducible representations of SL 2 (C) are irreducible representations of (SL 2 (C)) m . On the other hand, inner tensor products are not irreducible and their decomposition leads to a natural definition of the apolarity form (see Sect. 3.2). We now set out to compute these decompositions, which are often given as exercises in the literature (see, e.g., Exercise 11.11 of [6]).

A.1 Decomposition of V(n) ⊗ V(m)
Fix n, m ∈ N 0 . We now consider the representation of SL 2 (C) given by the inner tensor product, V (n)⊗V (m). The importance of the tensor product comes from the fact that it relates to consideration of SL 2 (C)-invariant bilinear forms like the apolarity form. In particular, the decomposition of the tensor product as a sum of irreducible representations (Proposition A.7) will show us exactly how the D map (see Proposition 3.1) can be used to define the apolarity form in the representation theoretic context (Definition 3.3).
We begin with an important SL 2 (C)-invariant map.

Proposition A.3 Let x and y denote the linear maps defined on V (k) via multiplication by x and y, respectively. The linear map U
Proof Straightforward computation, e.g., on simple tensors.
We then use this U map to show that D k is not the zero map when k ≤ m, n.
. We have: Proof Follows from the fact that (∂ x x − x∂ x ) p = p and (∂ y y − y∂ y ) p = p.
. We have: Proof Apply the previous lemma k times, and use the fact that D(x n−k ⊗ x m−k ) = 0.
We will use this fact about the image of D k to determine the decomposition of V (n)⊗V (m) into irreducible components. We will also need the following fundamental representation theory result. By Corollary A.5, this map is not the zero map, as x n−r · x m−r = x n+m−2r is in its image. Since V (n + m − 2r ) is irreducible by Proposition A.1, Schur's lemma implies this map is surjective. This in turn implies V (n) ⊗ V (m) ∼ = V (n + m − 2r ) + W r (as a representation) for some subspace W r . Since this holds for all r ≤ m, we actually have for some subspace W . The sum of irreducible components here is direct, as any two distinct irreducible components must intersect trivially. To show that we can set W = 0, we use the following dimension argument: Along with the stated decomposition, we also obtain something else: the r th transvectant is a projection from V (n) ⊗ V (m) onto the irreducible component V (n + m − 2r ). Schur's lemma and the tensor product decomposition then imply this projection is actually unique up to scalar. In a similar way, Schur's lemma also implies D and U must restrict to either a unique isomorphism or the zero map on each irreducible component of V (n) ⊗ V (m). In the following, we determine exactly what happens on each component. Proof By Schur's lemma, the claim immediately follows if U is injective and D is surjective. That D is surjective follows from the fact that the transvectant maps × • D r are projections onto each of the irreducible components of V (n)⊗V (m) for 1 ≤ r ≤ m+1. That U is injective follows from the fact that U (v) = 0 implies v = 0. One can see this by lexicographically ordering the basis {x j y n− j ⊗ x k y m−k : 0 ≤ j ≤ n, 0 ≤ k ≤ m} and considering the highest component of a given v ∈ V (n) ⊗ V (m).
Our main application of this theory is given as follows. Consider the nth transvectant map × • D n : V (n) ⊗ V (n) → V (0) ∼ = C, which is nonzero by the previous theorem. This map can be interpreted as an SL 2 (C)-invariant bilinear form on V (n). It turns out that the apolarity bilinear form used in Grace's theorem also has this property, and this justifies the following definition. Corollary A. 9 The apolarity form is the unique (up to scalar) nondegenerate SL 2 (C)-invariant bilinear form on V (n).

A.2 Decomposition of V( ) ⊗ V( )
Fix λ, μ ∈ N m 0 , and let V (λ) and V (μ) denote the irreducible representations of (SL 2 (C)) m given by the outer tensor products: We next generalize the above results to the inner tensor product of these two representations, V (λ)⊗ V (μ). In particular, we determine the decomposition of this tensor product and define a multivariate apolarity form. Note that these statements strictly generalize the previous analogous statements.
Proof We compute: The last step uses the distributive law for sums and tensor products of representations.
Proof Follows by induction on β, using Theorem A.8.

B The Grace-Walsh-Szegő coincidence theorem
A classical result in the representation theory of SL 2 (C) is the fact that V (n) ∼ = Sym n (V (1)). Here, Sym n (V (1)) denotes the set of symmetric tensors in V (1) ⊗n , or alternatively, the set of symmetric elements in V (1 n ). That is, there is some SL 2 (C)-invariant injection from V (n) to V (1) ⊗n , and by our conceptual thesis this map should transfer stability information. In fact, this idea is formalized in the Grace-Walsh-Szegő coincidence theorem, and the injective map is known as the polarization map.

B.1 Polarization and projection
For polynomials of degree m ≤ n, the degree-n polarization map is defined on monomials as follows and is extended linearly. This definition can be extended to homogeneous polynomials in V (n) by composing with Hmg −1 n and Hmg (1 n ) . The map ↑ n has a left inverse ↓ n , called the projection map, which we define as follows. (1 n ) and Hmg n . It is well-known that ↑ n is an injective linear map onto the subspace of symmetric multiaffine polynomials. This fact then extends to homogeneous polynomials, where the terms symmetric and multi-affine each refer to pairs of homogeneous variables. Further, one can define multivariate polarization and projection maps via composition: These two maps arise naturally in the theory of polynomials in general, and play an important role in the theory of stability, via the Grace-Walsh-Szegő coincidence theorem as well as in the proof of the Borcea-Brändén characterization of linear operators. The next result shows they also have represention theoretic importance. Proposition B.1 Fix λ ∈ N m 0 , and view V (λ) ∼ = V (λ 1 ) · · · V (λ m ) and V (1 λ ) ∼ = V (1) ⊗λ 1 · · · V (1) ⊗λ m as representations of (SL 2 (C)) m . The maps