Real Rank Geometry of Ternary Forms

We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we determine the real rank boundary: it is a hypersurface of degree 168. For quartics, sextics and septics we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants.

symmetric tensors of format 3×3× · · · ×3 with d factors. We are interested in decompositions where a i , b i , c i , λ i ∈ R for i = 1, 2, . . . , r . The smallest r for which such a representation exists is the real rank of f , denoted rk R ( f ). For d even, the representation (1) has signature (s, r − s), for s ≥ r/2, if s of the λ i 's are positive while the others are negative, or vice versa. The complex rank rk C ( f ) is the smallest r such that f has the form (1) where a i , b i , c i ∈ C.
The inequality rk C ( f ) ≤ rk R ( f ) always holds and is often strict. For binary forms, this phenomenon is well understood by now, thanks to [7,16]. For ternary forms, explicit regions where the inequality is strict were identified by Blekherman, Bernardi and Ottaviani in [6]. The present paper extends these studies. We focus on ternary forms f that are general in R[x, y, z] d . The complex rank of such a form is referred to as the generic rank. It depends only on d, and we denote it by R(d). The Alexander-Hirschowitz Theorem [2] implies that R(2) = 3, R(4) = 6, and R(d) = (d + 2)(d + 1) 6 otherwise.
We are particularly interested in general forms whose minimal decomposition is real. Set This is a full-dimensional semialgebraic subset of R[x, y, z] d . Its topological boundary ∂R d is the set-theoretic difference of the closure of R d minus the interior of the closure of R d .
Thus, if f ∈ ∂R d then every open neighborhood of f contains a general form of real rank equal to R(d) and also a general form of real rank bigger than R(d). The semialgebraic set ∂R d is either empty or pure of codimension 1. The real rank boundary, denoted ∂ alg (R d ), is defined as the Zariski closure of the topological boundary ∂R d in the complex projective space P(C[x, y, z] d ) = P ( d+2 2 )−1 . We conjecture that the variety ∂ alg (R d ) is non-empty and hence has codimension 1, for all d ≥ 4. This is equivalent to R(d) + 1 being a typical rank, in the sense of [6,7,16]. This is proved for d = 4, 5 in [6] and for d = 6, 7, 8 in this paper.
Our aim is to study these hypersurfaces. The big guiding problem is as follows: Problem 1. 1 Determine the polynomial that defines the real rank boundary ∂ alg (R d ).
The analogous question for binary forms was answered in [25,Theorem 4.1]. A related and equally difficult issue is to identify all the various open strata in the real rank stratification. This problem is open for d ≥ 4; the state of the art is the work of Bernardi, Blekherman and Ottaviani in [6]. For binary forms, this question has a complete answer, due to Blekherman [7], building on earlier work of Comon and Ottaviani [16]. See also [25, §4].
For any ternary form f and the generic rank r = R(d), it is natural to ask for the space of all decompositions (1). In the algebraic geometry literature [29,33], this space is denoted VSP( f ) and called the variety of sums of powers. By definition, VSP( f ) is the closure of the subscheme of the Hilbert scheme Hilb r (P 2 ) parametrizing the unordered configurations   20 Mukai [28], see also [33] Septics 12 5 points Dixon and Stuart [18] Octics 15 16 points Ranestad and Schreyer [33] that can occur in (1). If f is general then the dimension of its variety of sums of powers depends only on d. By counting parameters, the Alexander-Hirschowitz Theorem [2] implies In Table 1, we summarize what is known about the varieties of sums of powers. In two-thirds of all cases, the variety VSP( f ) is finite. It is one point only in the case of quintics, by [26]. We are interested in the semialgebraic subset SSP( f ) R of those configurations (3) in VSP( f ) whose r points all have real coordinates. This is the space of real sums of powers. Note that the space SSP( f ) R is non-empty if and only if the ternary form f lies in the semialgebraic set R d . The inclusion of SSP( f ) R in the real variety VSP( f ) R of real points of VSP( f ) is generally strict. Our aim is to describe these objects as explicitly as possible.
A key player is the apolar ideal of the form f . This is the 0-dimensional Gorenstein ideal A configuration (3) lies in VSP( f ) if and only if its homogeneous radical ideal is contained in f ⊥ . Hence, points in SSP( f ) R are 1-dimensional radical ideals in f ⊥ whose zeros are real. Another important tool is the middle catalecticant of f , which is defined as follows. For any partition d = u +v, consider the bilinear form C u,v ( f ) : R[x, y, z] u ×R[x, y, z] v → R that maps ( p, q) to the real number obtained by applying ( p · q)( ∂ ∂ x , ∂ ∂ y , ∂ ∂z ) to the polynomial f . We identify C u,v ( f ) with the matrix that represents the bilinear form with respect to the monomial basis. The middle catalecticant C( f ) of the ternary form f is precisely that matrix, where we take u = v = d/2 when d is even, and u =  (1) then C( f ) is the sum of the rank one matrices λ i C( d i ). If C( f ) has rank r then its signature is (# positive λ i , # negative λ i ). The identity SSP( f ) R = VSP( f ) R will be proved for d = 2 in Theorem 2.1, and the same argument works for d = 4, 6, 8 as well. The last assertion, for d ≤ 4, is due to Reznick [35,Theorem 4.6].
The structure of the paper is organized by increasing degrees: Section d is devoted to ternary forms of degree d. In Section 2, we determine the threefolds SSP( f ) R for quadrics, and in Section 3 we determine the surfaces SSP( f ) R for cubics. Theorem 3.1 summarizes the four cases displayed in Table 2. Section 4 is devoted to quartics f and their real rank boundaries. We present an algebraic characterization of SSP( f ) R as a subset of Mukai's Fano threefold V 22 , following [24,27,29,33,36]. In Section 5, we use the uniqueness of the rank 7 decomposition of quintics to determine the irreducible hypersurface ∂ alg (R 5 ). We also study the case of septics (d = 7), and we discuss VSP X for arbitrary varieties X ⊂ P N . Finally, Section 6 covers all we know about sextics, starting in Theorem 6.1 with a huge component of the boundary ∂ alg (R 6 ), and concluding with a case study of the monomial f = x 2 y 2 z 2 .
This paper contains numerous open problems and conjectures. We are fairly confident about some of them (like the one stated prior to Problem 1.1). However, others (like Conjectures 4.3 and 5.5) are based primarily on optimism. We hope that all will be useful in inspiring further progress on the real algebraic geometry of tensor decompositions.

Quadrics
The real rank geometry of quadratic forms is surprisingly delicate and interesting. Consider a general real quadric f in n variables. We know from linear algebra that rk R ( f ) = rk C ( f ) = n. More precisely, if ( p, q) is the signature of f then, after a linear change of coordinates, The stabilizer of f in GL(n, R) is denoted SO( p, q). It is called the indefinite special orthogonal group when p, q ≥ 1. We denote by SO + ( p, q) the connected component of SO( p, q) containing the identity. Let G denote the stabilizer in SO + ( p, q) of the set In particular, if f is positive definite then we get the group of rotations, SO + (n, 0) = SO(n), and G is the subgroup of rotational symmetries of the n-cube, which has order 2 n−1 n!.
Proof The analogue of the first assertation over an algebraically closed field appears in [34,Proposition 1.4]. To prove SSP( f ) R = SO + ( p, q)/G over R, we argue as follows. Every rank n decomposition of f has the form 2 j and is hence obtained from (6) by an invertible linear transformation x j → j that preserves f . These elements of GL(n, R) are taken up to sign reversals and permutations of the sets { 1 , . . . , p } and { p+1 , . . . , n }.
Suppose that f is not definite, i.e., p, q ≥ 1. Then we can write f = 2 2 1 − 2 2 2 + n j=3 ± 2 j . Over C, with i = √ −1, this can be rewritten as Table 2 Four possible types of a real cubic f of form (15) and its quadratic map F : There is an open set of such points.
Let f be definite and consider any point in VSP( f ) R . It corresponds to a decomposition where 1 , . . . , n are independent real linear forms and the a's and b's are in R. By rescaling 2 j and 2 j−1 , we obtain a 2 j−1 + ib 2 j−1 = 1. Adding the right-hand side to its complex conjugate, we get c j ∈ R and a 2 j + ib 2 j = 1. The catalecticant C( f ) is the matrix that represents f . A change of basis shows that C( f ) has ≥ k negative eigenvalues, hence k = 0.
The geometry of the inclusion SSP( f ) R into VSP( f ) R is already quite subtle in the case of binary forms, i.e., n = 2. We call f = a 0 x 2 + a 1 x y + a 2 y 2 hyperbolic if its signature is (1, 1). Otherwise f is definite. These two cases depend on the sign of the discriminant a 0 a 2 − 4a 2 1 .

Corollary 2.2 Let f be a binary quadric of rank 2. If f is definite then
Proof The apolar ideal f ⊥ is generated by two quadrics q 1 , The fibers of the map P 1 R → P 1 R given by (q 1 , q 2 ) consist of two points, corresponding to the decompositions f = 2 1 ± 2 2 . The fiber over (u : v) consists of the roots of the quadric uq 2 − vq 1 . If f is definite, then both roots are always real. Otherwise the discriminant with respect to (x, y), which is a quadric in (u, v), divides P 1 R into SSP( f ) R and its complement.

Example 2.3
Fix the hyperbolic quadric f = x 2 − y 2 . We take q 1 = x y and q 2 = x 2 + y 2 . The quadric uq 2 −vq 1 = u(x 2 +y 2 )−vx y has two real roots if and only if (2u −v)(2u +v) < 0. Hence SSP( f ) R is the interval in P 1 R defined by −1/2 < u/v < 1/2. In the topological description in Theorem 2.1, the group G is trivial, and SSP( f ) R is identified with the group

123
The homeomorphism between SO + (1, 1) and the interval between −1/2 and 1/2 is given by The resulting factorization u( It is instructive to examine the topology of the family of curves SSP( f ) R as f runs over the projective plane P 2 . This plane is divided by an oval into two regions: (i) the interior region {a 0 a 2 − 4a 2 1 < 0} is a disk, and it parametrizes the definite quadrics; (ii) the exterior region {a 0 a 2 −4a 2 1 > 0} is a Möbius strip, consisting of hyperbolic quadrics. Over the disk, the circles VSP( f ) R provide a trivial P 1 R -fibration. Over the Möbius strip, there is a twist. Namely, if we travel around the disk, along an S 1 in the Möbius strip, then the two endpoints of SSP( f ) R get switched. Hence, here we get the twisted circle bundle.
The topic of this paper is ternary forms, so we now fix n = 3. A real ternary form of rank 3 is either definite or hyperbolic. In the definite case, the normal form is f = x 2 + y 2 + z 2 , and SSP( f ) R = VSP( f ) R = SO(3)/G, where G has order 24. In the hyperbolic case, the normal form is f = x 2 + y 2 − z 2 , and SSP( f ) R VSP( f ) R = SO + (2, 1)/G, where G has order 4. These spaces are three-dimensional, and they sit inside the complex Fano threefold V 5 , as seen in Table 1. We follow [29,34] in developing our algebraic approach to SSP( f ) R . This sets the stage for our study of ternary forms of degree d ≥ 4 in the later sections.
Fix S = R[x, y, z] and f ∈ S 2 a quadric of rank 3. The apolar ideal f ⊥ ⊂ S is artinian, Gorenstein, and it has five quadratic generators. Its minimal free resolution has the form By the Buchsbaum-Eisenbud structure theorem, we can choose bases so that the matrix A is skew-symmetric. The entries are linear, so A = x A 1 + y A 2 + z A 3 where A 1 , A 2 , A 3 are real skew-symmetric 5×5-matrices. More invariantly, the matrices A 1 , The five quadratic generators of the apolar ideal f ⊥ are the 4×4-sub-Pfaffians of A.
The three points (a i : b i : c i ) in a decomposition (1) are defined by three of the five quadrics. Hence, VSP( f ) is identified with a subvariety of the Grassmannian Gr (3,5), defined by the condition that the three quadrics are the minors of a 2 × 3 matrix with linear entries. Equivalently, the chosen three quadrics need to have two linear syzygies. After taking a set of five minimal generators of f ⊥ containing three such quadrics, the matrix A has the form Here, 0 is the zero 2 × 2 matrix and T is a 3 × 2 matrix of linear forms. The 2 × 2-minors of T -which are also Pfaffians of A-are the three quadrics defining the points (a i : b i : c i ).

Proposition 2.4
The threefold VSP( f ) is the intersection of the Grassmannian Gr (3,5), in its Plücker embedding in P( 3 f ⊥ 3 ) P 9 , with the 6-dimensional linear subspace Proof This fact was first observed by Mukai [27]. See also [33, §1.5]. If U = u 1 ∧ u 2 ∧ u 3 lies in this intersection then the matrix A has the form (8) for any basis that contains u 1 , u 2 , u 3 .
Note that any general codimension 3 linear section of Gr (3,5) arises in this manner. In other words, we can start with three skew-symmetric 5 × 5-matrices A 1 , A 2 , A 3 and obtain VSP( f ) = Gr(3, 5) ∩ P 6 A for a unique quadratic form f . In algebraic geometry, this Fano threefold is denoted V 5 . It has degree 5 in P 9 and is known as the quintic del Pezzo threefold.
Our space SSP( f ) R is a semialgebraic subset of the real Fano threefold VSP( f ) R ⊂ P 9 R . If f is hyperbolic, then the inclusion is strict. We now extend Example 2.3 to this situation.
Example 2. 5 We shall compute the algebraic representation of SSP( f ) R for f = x 2 +y 2 −z 2 .
The apolar ideal f ⊥ is generated by the 4 × 4 Pfaffians of the skew-symmetric matrix Here e i j = e i ∧ e j . This is in the form (8). We fix affine coordinates on Gr(3, 5) as follows: If we write p i j for the signed 3 × 3-minors obtained by deleting columns i and j from this 3 × 5-matrix, then we see that VSP( f ) = P 6 A ∩ Gr (3,5) is defined by the affine equations p 12 − p 35 = ad − bc + e = 0, We now transform (10) into the coordinate system given by U and its orthogonal complement: The lower right 2 × 2-block is zero precisely when (12) holds. The upper right block equals Writing T = x T 1 + yT 2 + zT 3 , we regard T as a 2 × 3 × 3 tensor with slices T 1 , T 2 , T 3 whose entries are quadratic polynomials in a, b, c, d, e, g. The hyperdeterminant of that tensor equals In general, the expected degree of the hyperdeterminant of this form is 24. In this case, after some cancellations occur, this is a polynomial in 6 variables of degree 20 with 13956 terms. Now, consider any real point (a, b, c, d, e, g) that satisfies (12). The 2 ×2-minors of T define three points 1 , 2 , 3 in the complex projective plane P 2 . These three points are all real if and only if Det(T ) < 0.
Our derivation establishes the following result for the hyperbolic quadric f = x 2 + y 2 −z 2 . The solutions of (12) correspond to the decompositions f = 2 1 + 2 2 − 2 3 , as described above.

Corollary 2.6
In affine coordinates on the Grassmannian Gr (3,5), the real threefold VSP( f ) R is defined by the quadrics (12). The affine part of SSP( f ) R SO + (2, 1)/G is the semialgebraic subset of points (a, . . . , e, g) at which the hyperdeterminant Det(T ) is negative.
We close this section with an interpretation of hyperdeterminants (of next-to-boundary format) as Hurwitz forms [38]. This will be used in later sections to generalize Corollary 2.6. In this paper, we are concerned with the case n = 3. In Corollary 2.6 we took m = 2.

Cubics
The case d = 3 was studied by Banchi [4]. He gave a detailed analysis of the real ranks of ternary cubics f ∈ R[x, y, z] 3 with focus on the various special cases. In this section, we consider a general real cubic f . We shall prove the following result on its real decompositions.  Table 2. The algebraic boundary of SSP( f ) R is an irreducible sextic curve that has nine cusps.
Our point of departure is the following fact which is well known, e.g., from [4, §5] or [6].

Proposition 3.2
The real rank of a general ternary cubic is R(3) = 4, so it agrees with the complex rank. Hence, the closure of R 3 is all of R[x, y, z] 3 , and its boundary ∂R 3 is empty.
Proof Every smooth cubic curve in the complex projective plane P 2 can be transformed, by an invertible linear transformation τ ∈ PGL(3, C), into the Hesse normal form (cf. [3]): Suppose that the given cubic is defined over R. It is known classically that the matrix τ can be chosen to have real entries. In particular, the parameter λ will be real. Also, λ = −3; otherwise, the curve would be singular. Banchi [4] observed that 24(λ + 3) 2 f is equal to By applying τ −1 , one obtains the decomposition for the original cubic. The entries of the transformation matrix τ ∈ PGL(3, R) can be written in radicals in the coefficients of f . The corresponding Galois group is solvable and has order 432. It is the automorphism group of the Hesse pencil; see e.g., [3,Remark 4.2] or [15, Section 2]. (15) is well suited for this real structure. For any fixed isomorphism class of a real elliptic curve over C, there are two isomorphism classes over R, by [37, Proposition 2.2]. We see this by considering the j-invariant of the Hesse curve:

Remark 3.3 The Hesse normal form
For a fixed real value of j ( f ), this equation has two real solutions λ 1 and λ 2 . These two elliptic curves are isomorphic over C but not over R. They are distinguished by the sign of the degree 6 invariant T of ternary cubics, which takes the following value for the Hesse curve: If T ( f ) = 0 then the two curves differ by the sign of the Aronhold invariant. This proves that any real smooth cubic is isomorphic over R to exactly one element of the Hesse pencil. An illustrative example is the Fermat curve x 3 + y 3 + z 3 . It is unique over C, but it has two distinct real models, corresponding to λ = 0 or 6. The case λ = 6 is isomorphic over R to g = x 3 + (y + i z) 3 + (y − i z) 3 . This real cubic satisfies rk C (g) = 3 but rk R (g) = 4. Here, the real surface VSP(g) R is non-empty, but its semialgebraic subset SSP R (g) is empty.
We now construct the isomorphism VSP( f ) P 2 for ternary cubics f as shown in Table 1.
The apolar ideal f ⊥ is a complete intersection generated by three quadrics q 0 , q 1 , q 2 . We denote this net of quadrics by f ⊥ 2 . Conversely, any such complete intersection determines a unique cubic f . The linear system f ⊥ 2 defines a branched 4 : 1 covering of projective planes: We regard F as a map from P 2 to the Grassmannian Gr(2, f ⊥ 2 ) of 2-dimensional subspaces of f ⊥ 2 C 3 . It takes a point to the pencil of quadrics in f ⊥ 2 that vanish at . The fiber of F is the base locus of that pencil. Let B ⊂ P 2 be the branch locus of F. This is a curve of degree six. The fiber of F over any point in P 2 \B consists of four points 1 , 2 , 3 , 4 , and these determine decompositions f = 3 1 + 3 2 + 3 3 + 3 4 . In this manner, the rank 4 decompositions of f are in bijection with the points of P 2 \B. We conclude that 123 Second proof of Proposition 3. 2 We follow a geometric argument, due to De Paolis in 1886, as presented in [4, §5] and [6, §3]. Let H ( f ) be the Hessian of f , i.e., the 3 × 3 determinant of second partial derivatives of f . We choose a real line 1 that intersects the cubic H ( f ) in three distinct real points. The line 1 is identified with its defining linear form and hence with a point in the dual P 2 . That P 2 is the domain of F. We may assume that F( 1 ) is not in the branch locus B. There exists a decomposition f = 3 1 . We claim that the i have real coefficients. Let ∂ p ( f ) be the polar conic of f with respect to p = 1 ∩ 2 . This conic is a C-linear combination of 2 3 and 2 4 . It is singular at the point 3 ∩ 4 . In particular, p belongs to 1 and to the cubic H ( f ). Hence, p is a real point, the conic ∂ p ( f ) is real, and its singular point 3 ∩ 4 is real. The latter point is distinct from p = 1 ∩ 2 because f is smooth. After relabeling, all pairwise intersection points of the lines 1 , 2 , 3 , 4 are distinct and real. Hence the lines themselves are real.
The key step in the second proof is the choice of the line 1 . In practice, this is done by sampling linear forms 1 from R[x, y, z] 1 until H ( f ) ∩ 1 consists of three real points p. For each of these, we compute the singular point of the conic ∂ p ( f ) and connect it to p by a line. This gives the lines 2 , 3 , 4 ∈ R[x, y, z] 1 . The advantage of this method is that the coordinates of the i live in a cubic extension and are easy to express in terms of radicals.
In order to choose the initial line 1 more systematically, we must understand the structure of SSP( f ) R . This is our next topic. By definition, SSP( f ) R is the locus of real points p ∈ Example 3.4 Let f be the Hesse cubic (15). The net f ⊥ 2 is spanned by the three quadrics These quadrics define the map F : P 2 → P 2 . We use coordinates (x : y : z) on the domain P 2 and coordinates (a : b : c) on the image P 2 . The branch locus B of F is the sextic curve We regard the Hessian H ( f ) as a curve in the image P 2 . This cubic curve equals The ramification locus of the map F is the Jacobian of the net of quadrics: This cubic is known classically as the Cayleyan of f ; see So, we constructed four curves: the cubic C( f ) and the sextic H ( f ) ∨ in the domain P 2 = {(x : y : z)}, and the cubic H ( f ) and the sextic A smooth cubic f in the real projective plane is either a connected curve, namely a pseudoline, or it has two connected components, namely a pseudoline and an oval. In the latter case, f is hyperbolic. The cubic in the Hesse pencil (15) is singular for λ = −3, it is hyperbolic if λ < −3, and it is not hyperbolic if λ > −3. This trichotomy is the key for understanding SSP( f ) R . However, we must consider this trichotomy also for the Hessian cubic H ( f ) in (18) and for the Cayleyan cubic C( f ) in (19). The issue is whether their Hesse parameters − λ 3 +108 3λ 2 and 54−λ 3 9λ are bigger or smaller than the special value −3. The values at which the behavior changes are λ = −3, 0, 6. Table 2 summarizes the four possibilities.
Three possible hyperbolicity behaviors are exhibited by the three cubics f, H ( f ), C( f ). One of these behaviors leads to two different types, seen in the second and fourth column in Table 2. These two types are distinguished by the fibers of the map F : P 2 → P 2 . These fibers are classified by the connected components in the complement of the Cayleyan C( f ).
There are three such components if C( f ) is hyperbolic and two otherwise. The fifth row in Table 2 shows the number of real points over these components. For 6 < λ, there are no real points over one component; here, the general fibers have 4, 2 or 0 real points. However, for −3 < λ < 0, all fibers contain real points; here, the general fibers have 4, 2 or 4 real points.
Proof of Theorem 3.1 After a coordinate change by a matrix τ ∈ PGL(3, R), we can assume that the cubic f is in the Hesse pencil (15). Hence so are the associated cubics H ( f ) and C( f ). If we change the parameter λ so that all three cubics remain smooth, then the real topology of the map F is unchanged. This gives four different types for SSP( f ) R , the locus of fully real fibers. The sextic B divides the real projective plane into two or three connected components, depending on whether its dual cubic C( f ) = B ∨ is hyperbolic or not. Figures 1, 2, 3 and 4 illustrate the behavior of the map F in the four cases given by the columns in Table 2. Each figure shows the plane P 2 with coordinates (x : y : z) on the left and the plane P 2 with coordinates (a : b : c) on the right. The map F takes the left plane onto the right plane. The two planes are dual to each other. In particular, points on the left correspond to lines on the right. Each of the eight drawings shows a cubic curve and a sextic curve. The two curves on the left are dual to the two curves on the right.
In each right diagram, the thick red curve is the branch locus B and the thin blue curve is the Hessian H ( f ). In each left diagram, the turquoise curve is the Cayleyan C( f ) = B ∨ , and the thick blue curve is the sextic H ( f ) ∨ dual to the Hessian. Each of the eight cubics has either two or one connected components, depending on whether the curve is hyperbolic or not. The complement of the cubic in P 2 R has three or two connected components. The diagrams verify the hyperbolicity behavior stated in the third and fourth row of Table 2. Note that each sextic curve has the same number of components in P 2 R as its dual cubic. Consider the three cases where C( f ) is hyperbolic. These are in Figs. 1, 2 and 3. Here, P 2 R \B has three connected components. The fibers of F could have 0, 2 or 4 real points on these three regions. The innermost region has four real points in its fibers. It is bounded by the triangular connected component of the (red) branch curve B, which is dual to the pseudoline of C( f ). This innermost region is connected and contractible: it is a disk in P 2 R . If λ / ∈ [−3, 0] then this disk is exactly our set SSP( f ) R . This happens in Figs.1 and 3. However, the case λ ∈ (−3, 0) is different. This case is depicted in Fig. 2. Here, we see that SSP( f ) R consists of two regions. First, there is the disk as before, and second, we have the outermost region. This region is bounded by the oval that is shown as two unbounded branches on the right in Fig. 2. That region is homeomorphic to a Möbius strip in P 2 R . The key observation is that the fibers of F over that Möbius strip consist of four real points.  It remains to consider the case when C( f ) is not hyperbolic. This is shown in Fig. 4. The branch curve B = C( f ) ∨ divides P 2 R into two regions, one disk and one Möbius strip. The former corresponds to fibers with four real points, and the latter corresponds to fibers with two real points. We conclude that SSP( f ) R is a disk also in this last case. We might note, as a corollary, that all fibers of F : P 2 → P 2 contain real points, provided 0 < λ < 6.
For all four columns of Table 2, the algebraic boundary of the set SSP( f ) R is the branch curve B. This is a sextic with nine cusps because it is dual to the smooth cubic C( f ). The situation is even more interesting over the Möbius strip. We can continuously change the set { 1 , 2 , 3 , 4 }, reaching in the end the same as at the beginning, but cyclicly permuted.

Remark 3.5
Given a ternary cubic f with rational coefficients, how to decide whether SSP( f ) R has one or two connected components? The classification in Table 2 can be used for this task as follows. We first compute the j-invariant of f and then we substitute the rational number j ( f ) into (16). This gives a polynomial of degree 4 in the unknown λ. That polynomial has two distinct real roots λ 1 < λ 2 , provided j ( f ) / ∈ {0, 1728}. They satisfy λ 1 < λ 2 < −3, or 0 < λ 1 < λ 2 < 6, or (−3 < λ 1 < 0 and 6 < λ 2 ). Consider the involution that swaps λ 1 with λ 2 . This fixes the case in Fig. 1, and the case in Fig. 4, but it swaps the cases in Figs. 2 and 3. Thus this involution preserves the hyperbolicity behavior. We get two connected components, namely both the disk and Möbius strip, only in the last case. The correct λ is identified by comparing the sign of the degree six invariant T ( f ), as in (17).
Its j-invariant equals j ( f ) = 16384/5. The corresponding real Hesse curves have parameters λ 1 = −13.506 . . . and λ 2 = −5.57 . . ., so we are in the case of Fig. 1. Indeed, the curve V ( f ) is hyperbolic, as seen in [39, Figure 1]. Hence SSP( f ) R is a disk, shaped like a spiky triangle. The real decomposition above is right in its center. Moreover, we can check that T ( f ) < 0. Hence λ 1 provides the unique curve in the Hesse pencil that is isomorphic to f over R.  Table 2.
What happens for j ( f ) = 1728? Here, the two real forms of the complex curve V ( f ) differ: one is hyperbolic and the other one is not. For example, f 1 = x 3 − xz 2 − y 2 z is hyperbolic and f 2 = x 3 + xz 2 − y 2 z is not hyperbolic. These two cubics are isomorphic over C, with j ( f 1 ) = j ( f 2 ) = 1728, and they are also isomorphic to their Hessians and Cayleyans.
We find noteworthy that the topology of SSP( f ) R can distinguish between the two real forms of an elliptic curve. This happens when j Here the two real forms of the curve correspond to the second and fourth column in Table 2.
We close this section by explaining the last row of Table 2. It concerns the oriented matroid [5] of the configuration {(a i , b i , c i ) : i = 1, . . . , r } in the decompositions (1). For d = 3 the underlying matroid is always uniform. This is the content of the following lemma.

Lemma 3.8 Consider a ternary cubic f
i whose apolar ideal f ⊥ is generated by three quadrics. Then any three of the linear forms 1 , 2 , 3 , 4 are linearly independent.
Proof Suppose 1 , 2 , 3 are linearly dependent. They are annihilated by a linear operator p as in (5). Let q 1 and q 2 be independent linear operators that annihilate 4 . Then pq 1 and pq 2 are independent quadratic operators annihilating f . Adding a third quadric would not lead to a complete intersection. This is a contradiction, since f ⊥ is a complete intersection.
In the situation of Lemma 3.8, there is unique vector  Table 2 shows the resulting map from the five connected components to the three oriented matroids. Two of the fibers have cardinality one. For instance, the fiber over (+, +, −, −) is the Möbius strip in SSP( f ) R . This is the first of the following two cases.

ii) The Hessian H ( f ) is hyperbolic and the Cayleyan C( f ) is not hyperbolic if and only
if f is isomorphic to x 3 + y 3 + z 3 + (ax + by + cz) 3 where a, b, c are positive real numbers.

Remark 3.10
The fiber over the oriented matroid (+, +, +, +) consists of cubics of the form It may seem surprising that this space has three components in Table 2. One can pass from one component to another, for instance, by passing through singular cubics, like 24x yz

Quartics
In this section, we fix d = 4 and we consider a general ternary quartic f ∈ R[x, y, z] 4 . We have r = R(4) = 6 and dim(VSP( f )) = 3, so the quartic f admits a threefold of decompositions By the signature of f , we mean the signature of C( f ). This makes sense by Proposition 1.3. According to the Hilbert-Burch Theorem, the radical ideal I T of the point configuration 6 is generated by the 3×3-minors of a 4×3-matrix T = T 1 x +T 2 y +T 3 z, where T 1 , T 2 , T 3 ∈ R 4×3 . We interpret T also as a 3 × 3 × 4-tensor with entries in R, or as a 3 × 3-matrix of linear forms in 4 variables. The determinant of the latter matrix defines the cubic surface in P 3 that is the blow-up of the projective plane P 2 at the six points. The apolar ideal f ⊥ is generated by seven cubics, and I T ⊂ f ⊥ is generated by four of these.
Mukai [29] showed that VSP( f ) is a Fano threefold of type V 22 , and a more detailed study of this threefold was undertaken by Schreyer in [36]. The topology of the real points in that Fano threefold was studied by Kollár and Schreyer in [24]. Inside that real locus lives the semialgebraic set we are interested in. Namely, SSP( f ) R represents the set of radical ideals I T , arising from tensors T ∈ R 3×3×4 , such that I T ⊂ f ⊥ and the variety V (I T ) consists of six real points in P 2 . This is equivalent to saying that the cubic surface of T has 27 real lines.
Disregarding the condition I T ⊂ f ⊥ for the moment, this reality requirement defines a full-dimensional, connected semialgebraic region in the tensor space R 3×3×4 . The algebraic boundary of that region is defined by the hyperdeterminant Det(T ), which is an irreducible homogeneous polynomial of degree 48 in the 36 entries of T . Geometrically, this hyperdeterminant is the Hurwitz form in [38,Example 4.3]. This is Proposition 2.7 for m = n = 3.
We are interested in those ternary forms f whose apolar ideal f ⊥ contains I T for some T in the region described above. Namely, we wish to understand the semialgebraic set The following is a step toward understanding the algebraic boundary ∂ alg (R 4 ) of this set.

Theorem 4.1 The algebraic boundary ∂ alg (R 4 ) is a reducible hypersurface in the P 14 of quartics. One of its irreducible components has degree 51; that component divides the quartics of signature (5, 1). Another irreducible component divides the region of hyperbolic quartics.
Proof By [6, Example 4.6], ∂ alg (R 4 ) is non-empty, so it must be a hypersurface. We next identify the component of degree 51. The anti-polar of a quartic f , featured in [6, §5.1] and in [20], is defined by the following rank 1 update of the middle catalecticant: Writing "Adj" for the adjoint matrix, the Matrix Determinant Lemma implies ). Let f be a general quartic of signature (5, 1). Then det(C( f )) is negative, and the quartic ( f ) is non-singular. We claim that rk R ( f ) = 6 if and only if the curve ( f ) has a real point. The only-if direction is proved in a more general context in Lemma 6.4. For the if-direction, we note that the anti-polar quartic curve divides P 2 R into regions where ( f ) has opposite signs. Hence, we can find such that det C( f + 4 ) = 0, and therefore rk R ( f ) = 6. Examples in [6, §5.1] show that rk R ( f ) can be either 6 or 7. We conclude that, among quartics of signature (5, 1), the boundary of R 4 is given by the Blekherman discriminant Bdisc( f ) of degree 51.
To prove that ∂ alg (R 4 ) is reducible, we consider the following pencil of quartics: At t = 0, we obtain a quartic f 0 of signature (3, 3) that has real rank 6. One checks that f 0 is smooth and hyperbolic. We substitute f t into the invariant of degree 51 derived above, and we note that the resulting univariate polynomial in t has no positive real roots. So, the ray { f t } given by t ≥ 0 does not intersect the boundary component we already identified. For positive parameters t, the discriminant of f t is nonzero, until t reaches τ 1 = 6243.83 . . .. This means that f t is smooth hyperbolic for real parameters t between 0 and τ 1 . On the other hand, the rank of the middle catalecticant C( f t ) drops from 6 to 5 when t equals τ 0 = 3103.22 . . .. Hence, for τ 0 < t < τ 1 , the quartics f t are hyperbolic and of signature (4, 2). By [6, Corollary 4.8], these quartics have real rank at least 7. This means that the half-open interval given by (0, τ 0 ] crosses the boundary of R 4 in a new irreducible component.

Remark 4.2
One of the starting points of this project was the question whether rk R ( f ) ≥ 7 holds for all hyperbolic quartics f . This was shown to be false in [6,Remark 4.9]. The above quartic f 0 is an alternative counterexample, with an explicit rank 6 decomposition over Q.
We believe that, in the proof above, the crossing takes place at τ 0 , and that this newly discovered component is simply the determinant of the catalecticant det(C( f )). But we have not been able to certify this. Similar examples suggest that also the discriminant disc( f ) itself appears in the real rank boundary. Based on this, we propose the following conjecture.

Conjecture 4.3
The real rank boundary ∂ alg (R 4 ) for ternary quartics is a reducible hypersurface of degree 84 = 6 + 27 + 51 in P 14 . It has three irreducible components, namely the determinant of the catalecticant, the discriminant and the Blekherman discriminant. Algebraically, The construction of the Blekherman discriminant extends to the case when f is a sextic or octic; see Lemma 6.4. For quartics f , we can use it to prove rk R ( f ) > 6 also when the signature is (4, 2) or (3, 3). We illustrate this for the quartic given by four distinct lines.

Example 4.4
We claim that f = x yz(x + y + z) has rk R ( f ) = 7. For the upper bound, note The catalecticant C( f ) has signature (3, 3). The anti-polar quartic ( f ) = a 2 b 2 + a 2 c 2 + b 2 c 2 − a 2 bc − ab 2 c − abc 2 is nonnegative. By [6, Section 5.1], we have rk R ( f ) = 7. ♦ If f is a general ternary quartic of real rank 6, then SSP( f ) R is an open semialgebraic set inside the threefold VSP( f ) R . Our next goal is to derive an algebraic description of this set. We begin by reviewing some of the relevant algebraic geometry found in [17,27,29,33,36].
The Fano threefold VSP( f ) in its anti-canonical embedding is a subvariety of the Grassmannian Gr(4, 7) in its Plücker embedding in P 34 . It parametrizes 4-dimensional subspaces of f ⊥ 3 R 7 that can serve to span I T . In other words, the Fano threefold VSP( f ) represents quadruples of cubics in f ⊥ 3 that arise from a 3×3×4-tensor T as described above. Explicitly, VSP( f ) is the intersection of Gr(4, 7) with a linear subspace P 13 A in P 34 . This is analogous to Proposition 2.4, but more complicated. The resolution of the apolar ideal f ⊥ has the form By the Buchsbaum-Eisenbud structure theorem, we can write A = x A 1 + y A 2 + z A 3 where A 1 , A 2 , A 3 are real skew-symmetric 7×7-matrices. In other words, the matrices A 1 , A 2 , A 3 lie in 2 f ⊥ 3 . The seven cubic generators of the ideal f ⊥ are the 6×6-sub-Pfaffians of A. The ambient space P 34 for the Grassmannian Gr(4, 7) is the projectivation of the 35dimensional vector space 4 f ⊥ 3 . The matrices A 1 , A 2 , A 3 determine the following subspace: Each constraint U ∧ A i = 0 gives seven linear equations, for a total of 21 linear equations.

Lemma 4.5 The Fano threefold VSP( f ) of degree 22 is the intersection of the Grassmannian
Gr(4, 7) with the linear space P 13 A . Its defining ideal is generated by 45 quadrics, namely the 140 quadratic Plücker relations defining Gr(4, 7) modulo the 21 linear relations in (23).
Proof This description of the Fano threefold V 22 was given by Ranestad and Schreyer in [33] and by Dinew, Kapustka and Kapustka in [17,Section 2.3]. We verified the numbers 22 and 45 by a direct computation.
It is important to note that we can turn this construction around and start with any three general skew-symmetric 7 × 7-matrices A 1 , A 2 , A 3 . Then the 6 × 6-sub-Pfaffians of x A 1 + y A 2 + z A 3 generate a Gorenstein ideal whose socle generator is a ternary quartic f . This correspondence shows how to go from rank 6 decompositions of f to points U in VSP( f ) = Gr(4, 7) ∩ P 13 A ⊂ P 34 . Given the configuration X = {(a i : b i : c i )} in (20), the point U is the space of cubics that vanish on X. Conversely, given any point U in VSP( f ), we can choose a basis of f ⊥ 3 such that our 7 × 7-matrices A i have the form analogous to (8): Here T i is a 4 × 3-matrix, and 0 is the zero 3 × 3-matrix. The four 3 × 3-minors of the 4 × 3matrix T = x T 1 +yT 2 +zT 3 are among the seven 6×6-sub-Pfaffians of A = x A 1 +y A 2 +z A 3 . These four cubics define the six points in the decomposition (20).
We are now ready to extend the real geometry in Corollary 2.6 from quadrics to quartics. Let V = (v i j ) be a 4×3-matrix of unknowns. These serve as affine coordinates on Gr (4,7). Each point is the row span of the 4×7-matrix U = Id 4 V . This is analogous to (11).
Proceeding as in (13), we consider the skew-symmetric 7 × 7-matrix Its entries are linear forms in x, y, z whose coefficients are quadratic polynomials in the 12 affine coordinates v i j . Vanishing of the lower right 3 × 3-matrix defines VSP( f ). The matrix T is identified with a 4 × 3 × 3-tensor whose entries are quadratic polynomials in the v i j . The hyperdeterminant of format 4 × 3 × 3 has degree 48 in the tensor entries. For our tensor T , the entries are inhomogeneous polynomials of degree 2, so the degree of Det(T ) is bounded above by 96 = 2 × 48. A direct computation reveals that the actual degree is 84. The degree drop from 96 to 84 is analogous to the drop from 24 to 20 witnessed in (14).
At present, we do not know whether the hyperdeterminantal boundary always exists:
Our next tool for studying VSP( f ) is another quartic curve, derived from f , and endowed with a distinguished even theta characteristic θ . Recall that there is a unique (up to scaling) invariant of ternary cubics in degree 4. This is the Aronhold invariant, which vanishes on cubics g with rk C (g) ≤ 3. For the given quartic f , the Aronhold quartic S( f ) is defined by Following [19], we call f the Scorza quartic of S( f ). Points on S( f ) correspond to lines in the threefold VSP( f ). To see this, consider any decomposition f = 6 i=1 4 i , representing a point in VSP( f ). This point lies on a P 1 in VSP( f ) if and only if three of the lines 1 , 2 , . . . , 6 meet. Indeed, if a ∈ 1 ∩ 2 ∩ 3 in P 2 then ∂ p ( f ) is a sum of three cubes, i.e., p ∈ S( f ). We may regard 1 , 2 , 3 as linear forms in two variables, so that 4 1 + 4 2 + 4 3 is a binary quartic. This binary quartic has a P 1 of rank 4 decomposition, each giving a decomposition of f , with 4 , 5 , 6 fixed. The resulting line in VSP( f ) ⊂ P 13 A is the set of 4-planes U containing the P 3 of cubics Q · a, where Q is a quadric vanishing on 4 , 5 , 6 . This gives all lines on VSP( f ).
One approach we pursued is the relationship of the real rank of f with its topology in P 2 R . A classical result of Klein and Zeuthen, reviewed in [32,Theorem 1.7], states that there are six types of smooth plane quartics in P 2 R , and these types form connected subsets of P 14 R : 4 ovals, 3 ovals, 2 non-nested ovals, hyperbolic, 1 oval, empty curve. (26) We consider the pairs of types given by a general quartic f and its Aronhold quartic S( f ).

Proposition 4.8 Among the 36 pairs of topological types (26) of smooth quartic curves in the real projective plane P 2 R , at least 30 pairs are realized by a quartic f and its Aronhold quartic S( f ). Every pair not involving the hyperbolic type is realizable as f, S( f ) .
Proof This was established by exhaustive search. We generated random quartic curves using various sampling schemes, and this led to 30 types. The six missing types are listed in Conjecture 4.9. For a concrete example, here is an instance where f and S( f ) are empty: At the other end of the spectrum, let us consider For this quartic, both f and S( f ) have 28 real bitangents, so they consist of four ovals.

Conjecture 4.9 If a smooth quartic f on P 2 R is hyperbolic then its Aronhold quartic S( f ) is either empty or has two ovals. If f consists of three or four ovals then S( f ) is not hyperbolic.
We now describe the construction in [29] of eight distinguished Mukai decompositions The configuration 12 , 23 , 13 is a biscribed triangle of the Aronhold quartic S( f ). Being a biscribed triangle means that i j is tangent to S( f ) at a point q i j , the lines i j and ik meet at a point q i on the curve S( f ), and the line i is spanned by q i j and q ik .
We write VSP( f ) Muk for the subvariety of VSP( f ) given by Mukai decompositions (27). Mukai [29] showed that VSP( f ) Muk is a finite set with eight elements. We are interested in One idea we had for certifying rk R ( f ) = 6 is to compute the eight points in VSP( f ) Muk . If (27) is fully real for one of them then we are done. Unfortunately, this algorithm may fail. The semialgebraic set of quartics with real Mukai decompositions is strictly contained in R 4 :

Proposition 4.11
There exist quartics f of real rank 6 such that SSP( f ) Muk R is empty.
Proof Consider the eight Mukai decompositions (27) of the following quartic of real rank 6: The lines i and j in (27) intersect in the point q i j ∈ S( f ). If both lines are real then so is q i j . But, a computation shows that S( f ) has no real points. This implies SSP( f ) Muk R = ∅.
Example 4. 12 We close this section by discussing the Blum-Guinand quartics in [11]. Set The parameters a, b, m satisfy 0 < b < a and √ b/a < m < √ a/b. Blum-Guinand quartics have 28 real bitangents, so they consist of four ovals.

Quintics and septics
A general ternary quintic f ∈ R[x, y, z] 5 has complex rank R(5) = 7. The decomposition is unique by a classical result of Hilbert, Richmond and Palatini. Oeding and Ottaviani [30] explained how to compute the seven linear forms i by realizing them as eigenvectors of a certain 3×3×3-tensor. Inspired by [33, §1.5], we propose the following alternative algorithm: To prove the correctness of this algorithm, we recall what is known about the ideal J of seven points in P 2 . The ideal J is Cohen-Macaulay of codimension 2, so it is generated by the maximal minors of its Hilbert-Burch matrix T . According to [1,Theorem 5.1], this matrix has the following form if and only if no six of the seven points lie on a conic: 123 Here l 1 , l 2 , l 3 are independent linear forms and q 1 , q 2 , q 3 are quadratic forms in x, y, z.

Proposition 5.2 Algorithm 5.1 computes the unique decomposition of a general quintic f .
In the resulting representation (28), no six of the seven lines i are tangent to a conic.
Proof Let f be a general quintic. The apolar ideal f ⊥ in S is generated by four cubics g 1 , g 2 , g 3 , g 4 and one quartic h. This ideal is Gorenstein of codimension 3. The Buchsbaum-Eisenbud structure theorem implies that the minimal free resolution of f ⊥ has the form The matrix A is skew-symmetric of size 5 × 5, i.e., Here the l i 's are linear forms and the q i j 's are quadrics. As described above and in Section 2, we should find a 5 × 5 matrix U such that the lower right 2 × 2 submatrix of U · A · U t is the zero matrix. Since f is general, we may assume that the l i span R[x, y, z] 1  We perform row and column operations by the following right and left multiplication: The inverse column operation on the row vector B of minimal generators of f ⊥ gives Let J = g i − c i g 4 : i = 1, 2, 3 denote the ideal generated by the first three cubics. This is the ideal in Step 4 of the algorithm. We claim that V (J ) consists of seven points in P 2 .
By construction, we have B · A = 0, and the columns of A span the syzygies on B . The entries of B are the 4 × 4 sub-Pfaffians of A . The first three entries are the sub-Pfaffians that involve the last two rows and columns. These three 4 × 4 Pfaffians are the 2 × 2-minors of This is a Hilbert-Burch matrix for the ideal J . Hence J is an ideal of seven points in P 2 . Moreover, since l 1 , l 2 and l 3 are linearly independent, no six points of them lie on a conic. Dually, this means that no six of the seven lines i used in (28) are tangent to a conic.
It is easy to decide whether the real rank of a given ternary quintic f is 7 or not. Namely, one computes the unique complex decomposition (28) and checks whether it is real. The real rank boundary corresponds to transition points where two of the linear forms in (28) come together and form a complex conjugate pair. The following is our main result on quintics.

Theorem 5.3 The algebraic boundary
is an irreducible hypersurface of degree 168 in the P 20 of quintics. It has the parametric represen- Proof The parametrization (31) defines a unirational variety Y in P 20 . The Jacobian of this parametrization is found to have corank 1. This means that Y has codimension 1 in P 20 . Hence Y is an irreducible hypersurface, defined by a unique (up to sign) irreducible homogeneous polynomial in 21 unknowns, namely the coefficients of a ternary quintic. Let g be a real quintic (31) that is a general point in Y . For → 0, the real quintics ( 6 + 7 ) 5 − 5 6 and (i 6 + 7 ) 5 + (−i 6 + 7 ) 5 converge to the special quintic 4 6 7 in P 20 R . Hence any small neighborhood of g in P 20 R contains quintics of real rank 7 and quintics of real rank ≥ 8. This implies that Y lies in the algebraic boundary ∂ alg (R 5 ). Since Y is irreducible and has codimension 1, it follows that ∂ alg (R 5 ) exists and has Y as an irreducible component.
We carried out an explicit computation to determine that the (possibly reducible) hypersurface ∂ alg (R 5 ) has degree 168. This was done as follows. Fix the field K = Q(t), where t is a new variable. We picked random quintics f 1 and f 2 in Q[x, y, z] 5 , and we ran Algorithm 5.1 Step 4 returned a homogeneous ideal J in K [x, y, z] that defines 7 points in P 2 over the algebraic closure of K . By eliminating each of the three variables, we obtain binary forms of degree 7 in K [x, y], K [x, z] and K [y, z]. Their coefficients are polynomials of degree 35 in t. The discriminant of each binary form is a polynomial in Q[t] of degree 420 = 12 × 35. The greatest common divisor of these three discriminants is a polynomial (t) of degree 168. We checked that By definition, is an irreducible homogeneous polynomial with integer coefficients in the 21 coefficients of a general quintic f . Its specialization ( f 1 + t f 2 ) is a non-constant polynomial in Q[t], of degree deg(X ) in t. That polynomial divides (t). Since the latter is irreducible, we conclude that ( f 1 + t f 2 ) = γ · (t), where γ is a nonzero rational number. Hence has degree 168. We conclude that deg(Y ) = 168, and therefore Y = ∂ alg (R 5 ). Theorem 5.3 was stated for a very special situation, namely ternary quintics. We shall now describe a geometric generalization. Let X be any irreducible projective variety in the complex projective space P N that is defined over R and whose real points are Zariski dense. We set d = dim(X ). The generic rank is the smallest integer r such that the r th secant variety σ r (X ) equals P N . Given f ∈ P N , we define VSP X ( f ) to be the closure in the Hilbert scheme Hilb r (X ) of the set of configurations of r distinct points in X whose span contains f . Now, VSP stands for variety of sums of points. This object agrees with that studied by Gallet, Ranestad and Villamizar in [23]. It differs from more inclusive definitions seen in other articles. In particular, if N = d+2 2 − 1 and X = ν d (P 2 ) is the dth Veronese surface then VSP X ( f ) = VSP( f ). In this case, we recover the familiar variety of sums of powers.
The objects of real algebraic geometry studied in this paper generalize in a straightforward manner. We write VSP X ( f ) R for the variety of real points in VSP X ( f ), and we define SSP X ( f ) R to be the semialgebraic subset of all f that lie in an (r − 1)-plane spanned by r real points in X . Following Blekherman and Sinn [9], we are interested in generic points in P N R whose real rank equals the generic complex rank. These comprise the semialgebraic The topological boundary ∂R X is the closure of R X minus the interior of that closure. If X has more than one typical real rank, then ∂R X is non-empty and its Zariski closure ∂ alg (R X ) is a hypersurface in P N . This hypersurface is the real rank boundary we are interested in.
Example 5.4 Let N = 20 and X = ν 5 (P 2 ) the fifth Veronese surface in P 20 . Then r = 7 and ∂ alg (R X ) equals the irreducible hypersurface of degree 168 described in Theorem 5.3.
This example generalizes as follows. Let X ⊂ P N as above, and let τ (X ) denote its tangential variety. By definition, τ (X ) is the closure of the union of all lines that are tangent to X . We also consider the (r − 2)nd secant variety σ r −2 (X ). The expected dimensions are dim τ (X ) = 2d and dim σ r −2 (X ) = (r − 2)d + r − 3.
We are interested in the join of the two varieties, denoted σ r −2 (X ) τ (X ). This is an irreducible projective variety of expected dimension rd + r − 2 in P N . It comes with a distinguished parametrization, generalizing that in (31) for the Veronese surface of Example 5.4. The following generalization of Theorem 5.3 explains the geometry of the real rank boundary: One difficulty in proving this conjecture is that we do not know how to control interactions among the distinct decompositions f = f 1 +· · ·+ f r of a general point f ∈ P N into r points f 1 , . . . , f r on the variety X . Moreover, we do not even know that ∂ alg (R X ) is non-empty.
To illustrate Conjecture 5.5, we prove it in the case when X is the 7th Veronese surface. The parameters are d = 2, N = 35, and r = 12. We return to the previous notation, so f is a general ternary form in R[x, y, z] 7 . Here we can show that 13 is indeed a typical real rank.

Proposition 5.6
The real rank boundary ∂ alg (R 7 ) is a non-empty hypersurface in P 35 with one of the components equal to the join of the tenth secant variety and the tangential variety.
Proof For a general septic f , the minimal free resolution of the apolar ideal f ⊥ equals This is as in (7), but now the entries of the skew-symmetric 5 × 5-matrix A are quadratic. To find the five rank 12 decompositions of f , we proceed as in Example 2.5: we solve the matrix equation (13). The matrix entry in position (4, 5) is a homogeneous quadric in x, y, z whose six coefficients are inhomogeneous quadrics in the unknowns a, b, c, d, e, g. These coefficients must vanish. This system of six equations in six unknowns defines VSP( f ) in C 6 . It has precisely five solutions. For each of these solutions, we consider the upper right 3 × 2-matrix T . Its entries are quadrics in x, y, z. The 2 × 2-minors of T define the desired 12 points in P 2 .
We apply the above algorithm to the point in σ 10 (X ) τ(X ) given by the septic This septic f has complex rank 12, but its real rank is larger. The output of our decomposition algorithm shows that four of the five decompositions are not fully real. This remains true in a small neighborhood of f . Near the point (11x + 13y − 12z)(−18x + 13y − 16z) 6 in the tangential variety τ (X ), some septics have real rank 2 and some others have real rank 3. Hence, the above decomposition of f can change from purely real to a decomposition that contains complex linear forms. The same holds for all nearby points in the join variety. We conclude that a general point of the join in a small neighborhood of f belongs to ∂(R 7 ).
Our algorithm for septics f computes the five elements in VSP( f ) along with the 12 linear forms in each of the five decompositions f = 12 i=1 7 i . It outputs 60 points in P 2 . These come in 5 unlabeled groups of 12 unlabeled points in P 2 . Here are two concrete instances. i of real rank 12 that is defined by Here, all 60 points in P 2 are real. This means that the variety of sums of powers is fully real, and the twelve i in each of the five decompositions are real: Second, consider the real septic f = 6 i=1 ( 7 i +¯ 7 i ) defined by the complex linear forms This satisfies VSP( f ) R = SSP( f ) R = ∅. None of the 60 points in P 2 are real. Our two examples are extremal. One can easily find other septics f with VSP( f ) R = SSP( f ) R .

Sextics
We now consider ternary forms of degree six. The generic complex rank for sextics is R(6) = 10. Our first result states that both 10 and 11 are typical real ranks, in the sense of [6,7,16]. Theorem 6.1 The algebraic boundary ∂ alg (R 6 ) is a hypersurface in the P 27 of ternary sextics. One of its irreducible components is the dual to the Severi variety of rational sextics.
Proof We use notation and results from [8]. Let P 3,6 denote the convex cone of nonnegative rational sextics and 3,6 the subcone of sextics that are sums of squares of cubics over R. The dual cone ∨ 3,6 consists of sextics f whose middle catalecticant C( f ) is positive semidefinite. Its subcone P ∨ 3,6 is spanned by sixth powers of linear forms. It is known as the Veronese orbitope. The difference ∨ 3,6 \P ∨ 3,6 is a full-dimensional semialgebraic subset of R[x, y, z] 6 .
We claim that general sextics f in that set have real rank ≥ 11. Let f be a general sextic in ∨ 3,6 \P ∨ 3,6 . Suppose that rk R ( f ) = 10. The middle catalecticant C( f ) is positive definite. Proposition 1.3 tells us that the signature of any representation (1) is (10,0). This means that f lies in the Veronese orbitope P ∨ 3,6 . This is a contradiction to the hypothesis, and we conclude rk R ( f ) ≥ 11. Using [6, Theorem 1.1], this means that 11 is a typical rank.
Consider the algebraic boundary of the Veronese orbitope P ∨ 3,6 . One of its two components is the determinant of the catalecticant C( f ), which is the algebraic boundary of the spectrahedron ∨ 3,6 . The other component is the dual of the Zariski closure of the set of extreme rays of P 3,6 \ 3,6 . That Zariski closure was shown in [8,Theorem 2] to be equal to the Severi variety of rational sextics, which has codimension 10 and degree 26312976 in P 27 . Every generic boundary point of P ∨ 3,6 that is not in the spectrahedron ∨ 3,6 represents a linear functional whose maximum over P 3,6 occurs at a point in the Severi variety.
A result of Choi, Lam and Reznick (cf. [8,Proposition 7]) states that every general supporting hyperplane of P ∨ 3,6 touches the Veronese surface in precisely 10 rays. Every form in the cone spanned by these rays has real rank ≤ 10. Consider the subset of P ∨ 3,6 obtained by replacing each of the 10 rays by a small neighborhood. This defines a full-dimensional subset of forms f ∈ P ∨ 3,6 that satisfy rk R ( f ) = 10. By construction, this subset must intersect the boundary of P ∨ 3,6 in a relatively open set. Its Zariski closure is the hypersuface dual to the Severi variety. We conclude that this dual is an irreducible component of ∂ alg (R 6 ).

Remark 6.2
The same proof applies also for octics (d = 8), ensuring that the algebraic boundary ∂ alg (R 8 ) exists. Indeed, R(8) = 15 coincides with the size of the middle catalecticant f , and we can conclude that every octic in ∨ 3,8 \P ∨ 3,8 has real rank bigger than 15. However, for even integers d ≥ 10, this argument no longer works, because the generic complex rank exceeds the size of the middle catalecticant. In symbols, R(d) > d/2+2 2 . New ideas are needed to establish the existence of the hypersurface ∂ alg (R d ) for d ≥ 9.
We record the following upper bounds on the real ranks of general ternary forms. In particular, typical real ranks for ternary sextics are between 10 and 19.
Proof The same argument as in [ Let f ∈ R[x, y, z] 6 be general, with rk C ( f ) = R(6) = 10. In what follows we derive the algebraic representation of the K3 surface VSP( f ) and its semialgebraic subset SSP( f ) R .
Here A is a skew-symmetric 9×9-matrix with linear entries. Its 8×8 sub-Pfaffians generate f ⊥ . We write is a K3 surface of genus 20 and degree 38. See [28,33] for details and proofs. The following representation, found in [33, Theorem 1.7 (iii)], is analogous to Proposition 2.4 and Lemma 4.5.

Proposition 6.5
The surface VSP( f ) is the intersection of the Grassmannian Gr (5,9), in its Plücker embedding in P( 5 f ⊥ 4 ) P 125 , with the 20-dimensional linear subspace The upper right 5 × 4-submatrix of A drops rank precisely on the ten points (i : j : k) ∈ P 2 where i + j + k = 3 in nonnegative integers. We introduce local coordinates on Gr (5,9) as follows. Let U be the row span of Id 5 V , where V = (v i j ) is a 5 × 4 matrix of unknowns. We transform A into the coordinate system given by U and its orthogonal complement: We proceed as in Example 2.5. The lower right 4 × 4 block is zero whenever the corresponding 18 quadrics in the 20 local coordinates vanish. The submatrix T is a 3 × 4 × 5 tensor. Its hyperdeterminant Det(T ) is the discriminant for our problem. By Corollary 2.8, this is a polynomial of degree 120 in the entries of T . The specialization of Det(T ) to the 20 local coordinates has degree ≤ 240. That polynomial represents the algebraic boundary of SSP(f) R inside VSP( f ) R , similarly to Corollary 2.6.
At present we do not know whether the real rank of f is 11, 12 or 13. The argument above can be extended to establish the following result: if rk R ( f ) ≤ 12 then there exists a decomposition (1) whose points (a i : b i : c i ) all lie on the Fermat cubic V (x 3 + y 3 + z 3 ).
Let us explore the possibility rk R ( f ) = 12. Then it is likely that the (a i : b i : c i ) are the complete intersection of a cubic and a quartic. They can be assumed to have the form: x 3 + y 3 + 1 = ax 4 + bx 3 y + cx 3 + dx + ey + 1 = 0.
Hence determining the real rank of f leads directly to the following easy-to-state question: Can we find real constants a, b, c, d, e such that all 12 solutions to the equations (34) are real? If the answer is "yes" then we can conclude rk R ( f ) ≤ 12. Otherwise, we cannot reach a conclusion. A systematic approach to this real root classification problem is via the discriminant of the system (34). This discriminant is a polynomial of degree 24 in a, b, c, d, e.
We would need to explore the connected components of the complement of this hypersurface in R 5 . For further reading on the rank geometry of monomials, we refer to [12,13]. ♦