Coordinate-wise Powers of Algebraic Varieties

We introduce and study coordinate-wise powers of subvarieties of P, i.e. varieties arising from raising all points in a given subvariety of P to the r-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of P under the quotient of P by the action of the finite group Zn+1 r . We determine the degree of coordinate-wise powers and study their defining equations, particularly for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.


Introduction
Recently, Hadamard products of algebraic varieties have been attracting attention of geometers. These are subvarieties X Y of projective space P n that arise from multiplying coordinate-by-coordinate any two points x ∈ X, y ∈ Y in given subvarieties X, Y of P n . In applications, they first appeared in [CMS10], where the variety associated to the restricted Boltzmann machine was described as a repeated Hadamard product of the secant variety of P 1 × . . . × P 1 ⊂ P 2 n −1 with itself. Further study in [BCK16], [FOW17], [CCFL] made progress towards understanding Hadamard products. It seems natural to look at the r-th Hadamard power X r := X . . . X of an algebraic variety X and study the subvariety of X r given by coordinate-wise r-th powers of points in X ⊂ P n . Formally, for a projective variety X ⊂ P n and an integer r ∈ Z, we are interested in studying its image under the rational map ϕ r : P n P n , [x 0 : . . . : x n ] → [x r 0 : . . . : x r n ]. We denote the image of X under ϕ r by X •r and call it the r-th coordinate-wise power of X ⊂ P n .
In this article, we investigate these coordinate-wise powers X •r with a main focus on the case r > 0. These varieties show up naturally in many applications. For the Grassmannian variety Gr(k, P n ) in its Plücker embedding, the intersection with its r-th coordinate-wise power Gr(k, P n ) ∩ Gr(k, P n ) •r was described combinatorially in terms of matroids in [Len17] for even r. In [Bon18], highly singular surfaces in P 3 have been constructed as preimages of a specific singular surface under the morphism ϕ r for r > 0. The case r = −1 is the study of reciprocal varieties which has received particular attention in the case of linear spaces, see [DSV12], [KV16] and [FSW18].
For r > 0, the coordinate-wise powers X •r of a variety X ⊂ P n have the following natural interpretation: The quotient of P n by the finite subgroup Z n+1 r of the torus (C * ) n+1 is again a projective space. The image of a variety X ⊂ P n in P n /Z n+1 r ∼ = P n is the variety X •r , since ϕ r : P n → P n is the geometric quotient of P n by Z n+1 r . In other words, coordinate-wise powers of algebraic varieties are images of subvarieties of P n under the quotient by a certain finite group. The case r = 2 has the special geometric significance of quotienting by the group generated by reflections at the coordinate hyperplanes of P n . We are, therefore, especially interested in coordinate-wise squares of varieties.
A particular application of interest is the variety of orthostochastic matrices. An orthostochastic matrix is a matrix arising by squaring each entry of an orthogonal matrix. In other words, they are points in the coordinate-wise square of the variety of orthogonal matrices. Orthostochastic matrices play a central role in the theory of majorization [MOA11] and are closely linked to finding real symmetric matrices with prescribed eigenvalues and diagonal entries, see [Hor54] and [Mir63]. Recently, it has also been shown that studying the variety of orthostochastic matrices is central to the existence of determinantal representations of bivariate polynomials and their computations, see [Dey17a].
The article is structured as follows: As customary when studying any variety, first and foremost, we compute the degree of X •r . We use this to derive the degree of the variety of orthostochastic matrices. In Section 3, we dig a little deeper and find explicitly the defining equations of the coordinate-wise powers of hypersurfaces. We define generalised power sum hypersurfaces and give relations between their dual and reciprocal varieties.
We study in more detail coordinate-wise powers of linear spaces in the final section. We show the dependence of the degree of the coordinate-wise powers of a linear space on the combinatorial information captured by the corresponding linear matroid. Particular attention is drawn to the case of coordinate-wise squares of linear spaces. For lowdimensional linear spaces we give a complete classification. We also describe the defining ideal for the coordinate-wise square of general linear spaces of arbitrary dimension in a high-dimensional ambient space, and we link this question to the study of symmetric matrices with a codimension 1 eigenspace.
Acknowledgements. The authors would like to thank Mateusz Michałek and Bernd Sturmfels for their guidance and suggestions. This work was initiated while the first author was visiting Max Planck Institute MiS Leipzig. The financial support by MPI Leipzig which made this visit possible is gratefully acknowledged. The second and third author were funded by the International Max Planck Research School Mathematics in the Sciences (IMPRS) during this project.

Degree formula
Throughout this article, we work over C. We denote the homogeneous coordinate ring of P n by C[x] := C[x 0 , . . . , x n ]. For any integer r ∈ Z, we consider the rational map ϕ r : P n P n , [x 0 : . . . : x n ] → [x r 0 : . . . : x r n ]. For r ≥ 0, the rational map ϕ r is a morphism. Throughout, let X ⊂ P n be a projective variety, not necessarily irreducible. We denote by X •r ⊂ P n the image of X under the rational map ϕ r . More explicitly, if r ≥ 0.
For r < 0, we will only consider the case that no irreducible component of X is contained in any coordinate hyperplane of P n . We call the image X •r ⊂ P n the r-th coordinate-wise power of X. In the case r = −1, the variety X •(−1) is called the reciprocal variety of X. We primarily focus on positive coordinate-wise powers in this article, and therefore we will from now on always assume r > 0 unless explicitly stated otherwise. Observe that ϕ r : P n → P n is a finite morphism, and hence, the image X •r of X under ϕ r has same dimension as X.
The cyclic group Z r of order r is identified with the group of r-th roots of unity {ξ ∈ C | ξ r = 1}. We consider the action of the (n + 1)-fold product Z n+1 given by rescaling the variables x 0 , . . . , x n with r-th roots of unity. We denote the quotient of Z n+1 r by the subgroup {(ξ, ξ, . . . , ξ) ∈ C r | ξ r = 1} ⊂ Z n+1 r as G r := Z n+1 r /Z r . The group action of Z n+1 r on C[x] determines a linear action of G r on P n . In this way, we can also view G r as a subgroup of Aut(P n ). For r = 2, this has the geometric interpretation of being the linear group action generated by reflections at coordinate hyperplanes. Note that G r does not act on the vector space C[x] d of homogeneous polynomials of degree d, instead it acts on P(C[x] d ).
Given a projective variety, the following proposition describes the preimage under ϕ r of its coordinate-wise r-th power.
Proposition 2.1 (Preimages of coordinate-wise powers). Let X ⊂ P n be a variety and let X •r ⊂ P n be its coordinate-wise r-th power. The preimage ϕ −1 r (X •r ) is given by τ ∈Gr τ · X. Proof. This follows from X •r = ϕ r (X) and the fact that ϕ −1 r (ϕ r (p)) = {τ · p | τ ∈ G r } for all p ∈ X.
In particular, for r = 2, we obtain the following geometric description.
Corollary 2.2. The preimage of X •2 under ϕ 2 : P n → P n is the union over the orbit of X under the subgroup of Aut(P n ) generated by the reflections in the coordinate hyperplanes.
In the following theorem, we give a degree formula for the coordinate-wise powers of an irreducible variety. Theorem 2.3 (Degree formula). Let X ⊂ P n be an irreducible projective variety. Let Stab r (X) := {τ ∈ G r | τ · X = X} and Fix r (X) := {τ ∈ G r | τ | X = id X }. Then the degree of the r-th coordinate-wise power of X is Proof. Let H 1 , . . . , H k ⊂ P n for k := dim X •r = dim X be general hyperplanes whose common intersection with X •r consists of finitely many reduced points. We want to determine |X •r ∩ k i=1 H i |. By Proposition 2.1, we have For general hyperplanes H i , note that ϕ −1 r H i is a hypersurface of degree r and we have We note that Z := X ∩ τ ∈Gr\Stabr(X) τ · X is of dimension < k by irreducibility of X. Therefore, the common intersection of k general hyperplanes H i with ϕ r (Z) is empty, hence we can write the above as the following disjoint union: In particular, A general point in X •r = ϕ r (X) has |G r |/| Fix r (X)| preimages under ϕ r , so for general hyperplanes H i we conclude

Orthostochastic matrices
We use Theorem 2.3 to compute the degree of the variety of orthostochastic matrices. By O(m) ⊂ P m 2 (resp. SO(m) ⊂ P m 2 ) we mean the projective closure of the affine variety of orthogonal (resp. special orthogonal) matrices in A m 2 . It was shown in [Dey17a] that the problem of deciding whether a bivariate polynomial can be expressed as the determinant of a definite/monic symmetric linear matrix polynomial (a determinantal representation) is closely linked to the problem of finding the defining equations of the variety O(m) •2 . In the case m = 3, the defining equations of O(3) •2 are known [CÐ08, Proposition 3.1] and based on this knowledge, it was shown in [Dey17b, Section 4.2] how to compute a determinantal representation for a cubic bivariate polynomial or decide that none exists. For arbitrary m, the ideal of defining equations may be very complicated, but we are still able to compute its degree: Proof. The variety O(m) consists of two connected components that are isomorphic to SO(m). The images of these components under ϕ 2 : P m 2 → P m 2 coincide. In particular, O(m) •2 = SO(m) •2 and deg O(m) = 2 deg SO(m). We determine Fix 2 (SO(m)) and Stab 2 (SO(m)). Identify elements of G 2 with m × m-matrices whose entries are ±1. Then a group element S ∈ G 2 = {±1} m×m acts on the affine open subset A m 2 ⊂ P m 2 corresponding to m × m-matrices M ∈ C m×m as S • M , where S • M denotes the Hadamard product (i.e. entry-wise product) of matrices. Clearly, Fix 2 (SO(m)) is trivial, or else every special orthogonal matrix would need to have a zero entry at a certain position.
We claim that Stab 2 (SO(m)) ⊂ {S ∈ {±1} m×m | rk S = 1}. Indeed, assume that S ∈ {±1} m×m lies in Stab 2 (SO(m)), but is not of rank 1. Then m ≥ 2 and we may assume that the first two columns of S are linearly independent. Consider the vectors u, v ∈ C m given by Since u and v are orthogonal, we can find a special orthogonal matrix M ∈ C m×m whose first two columns are M •1 = u/ u 2 and M •2 = v/ v 2 . But S ∈ Stab 2 (SO(m)), so the matrix S • M must be a special orthogonal matrix. In particular, the first two columns of S • M must be orthogonal, i.e. (2. and equality in (2.1) holds if and only if S i1 S i2 = S j1 S j2 for all i, j ∈ {1, . . . , m}. However, this contradicts the linear independence of the first two columns of S. Hence, the claim follows.
Any rank 1 matrix in {±1} m×m can be uniquely written as uv T with u, v ∈ {±1} m and u 1 = 1. Such a rank 1 matrix S = uv T lies in Stab 2 (SO(m)) if and only if for each special orthogonal matrix M ∈ C m×m the matrix is again a special orthogonal matrix. This is true if and only if m and, thus, | Stab 2 (SO(m))| = 2 2m−2 .
Since SO(m) ⊂ P m 2 is irreducible, applying Theorem 2.3 gives .
Finally, we observe that the affine variety of orthogonal matrices in A m 2 is a complete intersection of m+1 2 quadrics which correspond to the polynomials given by the equation Remark 2.5. The degree of O(m) (resp. SO(m)) is known for all m by [BBBKR17], namely

Linear spaces
We now determine the degree of coordinate-wise powers L •r for a linear space L ⊂ P n , based on Theorem 2.3. It can be expressed in terms of the combinatorics captured by the matroid of L ⊂ P n . We briefly recall some basic definitions for matroids associated to linear spaces in P n . We refer to [Oxl11] for a detailed introduction to matroid theory.
Let L ⊂ P n be a linear space. The combinatorial information about the intersection of L with the linear coordinate spaces in P n is captured in the linear matroid M L of L. It is the collection of index sets I ⊂ {0, 1, . . . , n} such that L does not intersect V ({x i | i / ∈ I}). Formally, In the following result, we determine the degree of L •r ⊂ P n as an invariant of the linear matroid M L . Proof. By Theorem 2.3, we need to determine the cardinality of the groups Consider the affine cone over L, which is a (k + 1)-dimensional subspace W ⊂ C n+1 . We denote the canonical basis of C n+1 by e 0 , . . . , e n . We observe that | Fix r (L)| = |{τ ∈ Z n+1 From this, we see that | Fix r (L)| = r s . For the stabiliser of L, we have | Stab r (L)| = 1 r |{τ ∈ Z n+1 In particular, there are precisely r t elements τ ∈ Z n+1 r with τ · W = W . We deduce that | Stab r (L)| = r t−1 , which concludes the proof by Theorem 2.3.
Corollary 2.7. The degree of the coordinate-wise r-th power of a linear space only depends on the associated linear matroid. If L 1 , L 2 ⊂ P n are linear spaces such that the linear matroids M L 1 and M L 2 are isomorphic (i.e. they only differ by a permutation of {0, 1, . . . , n}), then L •r 1 ⊂ P n and L •r 2 ⊂ P n have the same degree. Corollary 2.8. Let L ⊂ P n be a linear space of dimension k. Then deg L •r ≤ r k . For general k-dimensional linear spaces in P n , equality holds.
Proof. Every coloop of M L forms a component of M L and the set {0, 1, . . . , n}\{coloops} is a union of components, hence t ≤ s + 1. Therefore, by Proposition 2.6, deg L •r ≤ r k . For a general linear space L ∈ Gr(k, P n ), the linear matroid M L has no coloops and only one component.

Hypersurfaces
In this section, we study the coordinate-wise powers of hypersurfaces. Here, by a hypersurface, we mean a pure codimension 1 variety. In particular, hypersurfaces are assumed to be reduced, but are allowed to have multiple irreducible components. We describe a way to find the explicit equation describing the image of the given hypersurface under the morphism ϕ r . We define generalised power sum symmetric polynomials and we give a relation between duality and reciprocity of hypersurfaces defined by them. Finally, we raise the question whether and how the explicit description of coordinate-wise powers of hypersurfaces may lead to results on the coordinate-wise powers for arbitrary varieties.

The defining equation
The defining equation of a degree d hypersurface is a square-free (i.e. reduced) polynomial unique up to scaling, corresponding to a unique f ∈ P(C[x] d ). We work with points in P(C[x] d ), i.e. polynomials up to scaling. We do not always make explicit which degree d we are talking about if it is irrelevant to the discussion. The product of f ∈ P(C[x] d ) and g ∈ P(C[x] d ) is well-defined up to scaling, i.e. as an element f g ∈ P(C[x] d+d ). Equally, we talk about irreducible factors etc. of elements of P(C[x] d ).
Since the finite morphism ϕ r preserves dimensions, the coordinate-wise r-th power of a hypersurface is again a hypersurface, leading to the following definition.
For a given square-free polynomial f , we want to compute f •r . To this end, we introduce the following auxiliary notion.
Observe that in case (ii), determining s r (f ) = lcm{s r (f 1 ), s r (f 2 ), . . . , s r (f m )} is straightforward, assuming the decomposition of f into irreducible factors f 1 , . . . , f m is known. Indeed, the irreducible factors of each s r (f i ) are immediate from case (i) of the definition, so determining the least common multiple does not require any additional factorization.
Proof. It is enough to show the claim for f irreducible because we can deduce the general case in the following manner. If f factors into irreducible factors as We now assume that f is irreducible. If f = x i for some i ∈ {0, 1, . . . , n}, then the claim holds trivially by the definition of s r (f ). Let f = x i for all i and g be a polynomial representing s r (f ) ∈ P(C[x] md ). By definition, s r (f ) is fixed under the action of G r , hence τ · g is a multiple of g for all τ ∈ Z n+1 r . Since g is not divisible by x i , it must contain a monomial not divisible by x i . This shows that g is fixed by . Based on Definition 3.2 and Lemma 3.3, the following proposition gives a method to find the equation of the coordinate-wise power of a hypersurface.

Proposition 3.4 (Powers of hypersurfaces).
Let V (f ) ⊂ P n be a hypersurface. The defining equation f •r of its coordinate-wise r-th power is given by replacing each occurrence of is the preimage under the ring homomorphism ψ : The claim is therefore an immediate consequence of Lemma 3.3.
For clarity, we illustrate the above results for a hyperplane in P 3 .
Example 3.5. For n = 3 and f := Expanding this expression, we obtain a polynomial in C[x 2 0 , x 2 1 , x 2 2 , x 2 3 ] and, substituting This rational quartic surface is illustrated in Figure 3.1. It is a Steiner surface with three singular lines forming the ramification locus of Example 3.6 (Squaring the circle). Consider the plane conic C = V (f ) ⊂ P 2 given by In the affine chart x 0 = 1, this corresponds over the real numbers to the circle with center (a, b) and radius c. From Proposition 3.4, we now see that the coordinate-wise square of the circle C ⊂ P 2 can be a line, a parabola or a singular quartic curve. See Figure 3.2 for an illustration of the following three cases: If the center of the circle lies on a coordinate-axis and is not the origin (i.e. ab = 0, but (a, b) = (0, 0)), then C •2 ⊂ P 2 is a conic. Say a = 0, then C •2 is defined by the equation In the affine chart x 0 = 1, C is a circle and C •2 is a parabola. (iii) If the center of the circle does not lie on a coordinate-axis, then |G r · f | = 4. Therefore, C •2 is a quartic plane curve. Its equation can be computed explicitly using Proposition 3.4. Being the image of a conic, the quartic curve C •2 is rational, hence it cannot be smooth. In fact, its singularities are the two points [0 : 1 : Then the Newton polytope of f •r arises from the Newton polytope of f by rescaling according to the cardinality of the orbit : Indeed, we have Newt(τ · f ) = Newt(f ) for all τ ∈ G r , and since Newt(gh) = Newt(g) + Newt(h) holds for all polynomials g, h, we have Newt(s r (f )) = |G r · f | · Newt(f ) by Definition 3.2. Replacing x r i by x i rescales the Newton polytope with the factor 1 r , so the claim follows.

Duals and reciprocals of power sum hypersurfaces
We now highlight the interactions between coordinate-wise powers, dual and reciprocal varieties for the case of power sum hypersurfaces V ( determine explicitly all hypersurfaces that arise from power sum hypersurfaces by repeatedly taking duals and reciprocals as the coordinate-wise r-th power of some hypersurface.
In this subsection, we also allow r to take negative integer values.
Recall that the reciprocal variety V (f ) •(−1) of a hypersurface V (f ) ⊂ P n not containing any coordinate hyperplane of P n is defined as the closure of ϕ −1 (V (f ) \ V (x 0 x 1 . . . x n )) in P n . We denote it also by R V (f ). For linear spaces the reciprocal variety and its Chow form has been studied in detail in [KV16].
We also recall the definition of the dual variety of V (f ) ⊂ P n . Consider the set of hyperplanes in P n that arise as the projective tangent space at a smooth point of V (f ). This is a subset of the dual projective space (P n ) * and its Zariski closure is the dual variety of V (f ), which we denote by V (f ) * or D V (f ). We identify (C n+1 ) * with C n+1 via the standard bilinear form and therefore identify (P n ) * with P n .
Consider the power sum polynomial As before, we regard polynomials only up to scaling. For power sums with negative exponents we consider the numerator of the rational function as In particular, f −1 ∈ P(C[x] n ) is the elementary symmetric polynomial of degree n.
Recall that the morphism ϕ r : P n → P n for r > 0 is finite, hence preserves dimension. Since ϕ −1 : P n P n is a birational map, the rational map where we have used the surjectivity of ϕ λ : P n → P n . For λ < 0, we use the above to see This naturally leads us to the our next definition. Definition 3.9 (Generalised power sum polynomial). For any rational number p = s r ∈ Q (r, s ∈ Z, r = 0), we define the generalised power sum polynomial f p := f •r s ∈ P(C[x] d ). By Lemma 3.8, the generalised power sum polynomial f p is well-defined. With this definition, we get the following duality result for hypersurfaces generalising Example 4.16 in [GKZ94]. It is an algebraic incarnation of the duality theory for p -spaces.
Proposition 3.10 (Duality of generalised power sum hypersurfaces). Let p, q ∈ Q \ {0} be such that 1 . . x n ) induces a linear isomorphism on projective tangent spaces T a P n = P n → P n = T b P n given by diag(ra r−1 0 , ra r−1 1 , . . . , ra r−1 n ). This maps In particular, V (f p ) * ⊂ P n is the image of the rational map : Using Proposition 3.4 we can compute f p for any p ∈ Q explicitly. In particular, we make the following observation: Lemma 3.11. Let s ∈ N and r ∈ Z be relatively prime. Then f s/r arises from f 1/r by substituting Proof. This follows from the explicit description of the polynomials f s/r = f •r s and f 1/r = f •r 1 given by Proposition 3.4. By Lemma 3.11, in order to determine the generalised power sum polynomials f p , we may restrict our attention to f 1/r . These have a particular geometric interpretation as repeated dual-reciprocals of the linear space V (x 0 + x 1 + . . . + x n ) ⊂ P n .
Theorem 3.12. For r > 0, the repeated alternating reciprocals and duals of the linear space V (f 1 ) ⊂ P n are the coordinate-wise powers of V (f 1 ) given as Proof. We show the claim by induction on r, with the case r = 1 amounting to the observation R V (f 1 ) = V (f −1 ). For r > 1, we get by induction hypothesis, where ( * ) follows from Lemma 3.8 and ( * * ) from Proposition 3.10. Taking the reciprocal variety of both sides gives the other identity Example 3.13. Let n = 3 and f := x 0 + x 1 + x 2 + x 3 . The reciprocal variety of the plane V (f ) ⊂ P 3 is given by

From hypersurfaces to arbitrary varieties?
We briefly discuss to what extent Proposition 3.4 can be used to determine coordinatewise powers of arbitrary varieties, and mention the difficulties involved in this approach.
If f 1 , . . . , f m are homogeneous polynomials vanishing on a variety X ⊂ P n , then their coordinate-wise powers give rise to the inclusion X •r ⊂ V (f •r 1 , . . . , f •r m ). We may ask when equality holds, which leads us to the following definition, reminiscent of the notion of tropical bases in Tropical Geometry [MS15, Section 2.6].
. We show the existence of such power bases for a given ideal in the following proposition.  Enlarging f 1 , . . . , f m to a generating set of I gives an r-th power basis of I.
Proposition 3.15 shows the existence of r-th power bases, but explicitly determining one a priori is nontrivial. In the following two examples, we will see that even in the case of squaring codimension 2 linear spaces, obvious candidates for f 1 , . . . , f m do not form a power basis.
Example 3.16. Let I := (f 1 , f 2 ) ⊂ C[x] be the ideal defining the line in P 3 that is given by f 1 := x 0 + x 1 + x 2 + x 3 and f 2 := x 1 + 2x 2 + 3x 3 . The polynomials f •2 1 and f •2 2 have degrees 4 and 2, respectively, by Proposition 3.4. Note that the polynomial x 2 )f 2 also lies in I, so the ideal of V (I) •2 contains the linear form f •2 3 = 3x 0 − x 1 + x 2 − 3x 3 . The polynomials f 1 , f 2 do not form a power basis of I. In fact, one can check that V (f •2 1 , f •2 2 ) ⊂ P 3 is the union of four rational quadratic curves, one of which is V (I) •2 , see Figure 3.4 for an illustration. A power basis of I is given by f 1 , f 2 , f 3 . . . , f m in the ideal of a linear space X ⊂ P n consists of the circuit forms, i.e. linear forms vanishing on X that are minimal with respect to the set of occurring variables. However, for these circuit forms are and one can check that the point [16 : 16 : 1 : 36 : 9] ∈ P 4 lies in V (f •2 1 , . . . , f •2 5 ), but not in X •2 . In particular, f 1 , . . . , f 5 is not an r-th power basis for r = 2.
We have seen in Example 3.16 and Example 3.17 that even for the case of linear spaces of codimension 2 it is not an easy task to a priori identify an r-th power basis.
The following proposition shows how one can straightforwardly find a very large r-th power basis of an ideal I, without first computing the ideal of V (I) •r .
In particular, Proposition 3.18 shows that for a subvariety of P n defined by k forms of degree d, its coordinate-wise r-th power can be described set-theoretically by the vanishing of (k − 1)r n + 1 forms of degree ≤ dr n−1 . However, we will see in Section 4 that for linear spaces this bound is rather weak in many cases and should be expected to allow dramatic refinement in general. We raise the following as a broad open question: I, can we find f 1 , . . . , f m ∈ I simultaneously forming an r-th power basis for all r?

Linear spaces
In this section, we specialise to linear spaces L ⊂ P n and study their coordinate-wise powers L •r . First, we highlight the dependence of L •r on the geometry of a finite point configuration associated to L ⊂ P n . Based on this, we classify the coordinate-wise squares of lines and planes. Finally, we turn to the case of squaring linear spaces in high ambient space.

Point configurations
We investigate the defining ideal of L •r for a linear space L ⊂ P n . The degrees of its minimal generators do not change under rescaling and permuting coordinates of P n , i.e. under the actions of the algebraic torus G n+1 m = (C * ) n+1 and the symmetric group S n+1 . Fixing a (k + 1)-dimensional vector space W , we have the identification Hence, we may express coordinate-wise powers of a linear space L in terms of the corresponding finite multi-set Z ⊂ PW * . In fact, it is easy to check that the degrees of the minimal generators of the defining ideal only depend on the underlying set Z, forgetting repetitions in the multi-set. We study coordinate-wise powers of a linear space in terms of the corresponding non-degenerate finite point configuration.
For the entirety of Section 4, we establish the following notation: Let L ⊂ P n be a linear space of dimension k. We understand L as the image of a chosen linear embedding ι : PW −−−−→ P n , where W is a (k + 1)-dimensional vector space and 0 , . . . , n ∈ W * are linear forms defining ι. Consider the finite set of points Z ⊂ PW * given by Since 0 , 1 , . . . , n ∈ W * define the linear embedding ι, they cannot have a common zero in W . Hence, the linear span of Z is the whole space PW * . We denote by I(Z) ⊂ Sym • W the defining ideal of Z ⊂ PW * . The subspace of degree r forms vanishing on Z is written as I(Z) r ⊂ Sym r W .
The main technical tool is the following observation that L •r ⊂ P n equals (up to a linear re-embedding) the image of the r-th Veronese variety ν r (PW ) ⊂ P Sym r W under the projection from the linear space P(I(Z) r ) ⊂ P Sym r W . Proof. We observe that the morphism ϕ r • ι is given by The n + 1 elements r i ∈ Sym r W * correspond to a linear map χ : Sym r W → C n+1 via the natural identification (Sym r W * ) n+1 = Hom C (Sym r W, C n+1 ).
Let f ∈ Sym r W such that f ∈ I(Z) r . Naturally identifying W and W * * , we may view f as a form of degree r on W * . Then, the condition that f ∈ I(Z) r translates to f ( i ) = 0 ∀i. Viewing f as a symmetric r-linear form W * × . . . × W * → C, we have f ( i , . . . , i ) = 0 ∀i. Also, when f is considered as a linear form on Sym r W * , f ( r i ) = 0 ∀i. The latter expression is equivalent to f ∈ ker χ, via the identification of W and W * * . We conclude I(Z) r = ker χ.
In particular, we deduce the following: Proof. Since I(Z) r = 0, we deduce from Lemma 4.1 that L •r = ϕ r (L) is a linear reembedding of the k-dimensional r-th Veronese variety ν r (PW ) ⊂ P Sym r W . The ideal of this Veronese variety is generated by quadrics. Since dim Sym r W = k+r r , the linear re-embedding ϑ : P Sym r W → P n adds n − k+r r + 1 linear forms to the ideal.

Squaring lines and planes
We now specialise to the case of coordinate-wise squaring, i.e. r = 2. This case has special geometric importance, since it corresponds to computing the image of a linear space under the quotient of P n by the reflection group generated by the coordinate hyperplanes.
We start out with low-dimensional cases and classify the coordinate-wise squares of lines and planes in arbitrary ambient spaces. Proof. Since Z ⊂ PW * spans the projective line PW * , we must have |Z| ≥ 2. If |Z| > 2, then I(Z) 2 = 0, since no non-zero quadratic form on the projective line PW * vanishes on all points of Z. Then Lemma 4.1 implies that L •2 = (ϕ 2 • ι)(PW ) is a linear re-embedding of ν 2 (PW ), which is a smooth conic in the plane P Sym 2 W ∼ = P 2 .
If |Z| = 2, then dim I(Z) 2 = 1, since up to scaling there is a unique quadric vanishing on the points Z. By Lemma 4.1, the image ϕ 2 (L) lies in a projective line P 1 ∼ = ϑ(P(Sym 2 W/I(Z) 2 )) ⊂ P n . On the other hand dim L •2 = dim L = 1. Hence, L •2 = ϕ 2 (L) is a line in P n .
Remark 4.4. We observe that the two possibilities in Theorem 4.3 for the coordinatewise square of a line L differ in degree. In particular, Corollary 2.7 shows that it only depends on the linear matroid M L whether L •2 is a line or a (re-embedded) plane conic.
Remark 4.5. In the Grassmannian of lines Gr(1, P n ), consider the locus Γ ⊂ Gr(1, P n ) of those lines L whose coordinate-wise square L •2 is a line. Considering Plücker coordinates p ij on the Grassmannian Gr(1, P n ), we observe that Γ is the subvariety of Gr(1, P n ) given by the vanishing of p ij p jk p ki for all i, j, k ∈ {0, 1, . . . , n} distinct: Indeed, if L is the image of an embedding P 1 B − → P n given by a chosen rank 2 matrix B ∈ C (n+1)×2 , then Z ⊂ (P 1 ) * is the set of points corresponding to the non-zero rows of B. Then |Z| = 2 if and only if among any three distinct rows of B there always exist two linearly dependent rows. In terms of the Plücker coordinates, which are given by the 2 × 2-minors of B, this translates into the vanishing condition above.  Proof. Notice that k = 2, so dim W = 3. (i) If I(Z) 2 = 0, then L •2 ⊂ P n is by Lemma 4.1 a linear re-embedding of the Veronese surface ν 2 (PW ) ⊂ P Sym 2 W . The ideal of the ν 2 (PW ) is minimally generated by six quadrics. Indeed, choosing a basis for W , we may understand points in P Sym 2 W as symmetric 3×3-matrices up to scaling. Then ν 2 (PW ) is the subvariety corresponding to symmetric rank 1 matrices, which is cut out by the six quadratic polynomials imposing vanishing 2 × 2-minors. Since dim P Sym 2 W = 5, the linear re-embedding P Sym 2 W → P n adds n − 5 linear forms to I. (ii) We can choose a basis {z 0 , z 1 , z 2 } of W such that the unique reduced plane conic through Z ⊂ PW * is with respect to these coordinates given by the vanishing of either q 1 := z 2 0 − 2z 1 z 2 ∈ Sym 2 W or q 2 := z 1 z 2 ∈ Sym 2 W . We consider the basis {z 2 1 , z 2 2 , 2z 0 z 1 , 2z 0 z 2 , 2z 1 z 2 } of Sym 2 W/ q 1 and the basis {z 2 0 , z 2 1 , z 2 2 , 2z 0 z 1 , 2z 0 z 2 } of Sym 2 W/ q 2 . With respect to these choices of bases, the morphism ψ : PW → P(Sym 2 W/I(Z) 2 ) is given as ψ : P 2 → P 4 , [a 0 : a 1 : a 2 ] → [a 2 1 : a 2 2 : a 0 a 1 : a 0 a 2 : a 2 0 + a 1 a 2 ] or ψ : P 2 → P 4 , [a 0 : a 1 : a 2 ] → [a 2 0 : a 2 1 : a 2 2 : a 0 a 1 : a 0 a 2 ]. In the first case, we checked computationally with Macaulay2 [GS] that the ideal is minimally generated by seven cubics. A structural description of these quadrics and cubics will be given in the proof of Theorem 4.9. The image of the second morphism is a complete intersection of two binomial quadrics. By Lemma 4.1, the coordinate-wise square L •2 arises from the image of ψ via a linear re-embedding P 4 → P n , producing additional n − 4 linear forms in I. (iii) In case (a), the set Z consists of three points spanning the projective plane PW * , so dim Sym 2 W/I(Z) 2 = 3. Then by Lemma 4.1, the coordinate-wise square L •2 is contained in a plane P 2 ∼ = ϑ(P(Sym 2 W/I(Z) 2 )) ⊂ P n . On the other hand, dim L •2 = dim L = 2, so L •2 ⊂ P n must be a plane in P n . For case (b), we may assume that , so Z can also be viewed as the finite set of points associated to L . Applying Lemma 4.1 to L ⊂ P 3 shows that the image of ψ : PW → P(Sym 2 W/I(Z) 2 ) is the coordinatewise square L •2 ⊂ P 3 . Hence, L •2 ⊂ P n is a linear re-embedding of the quartic surface from Example 3.5 into higher dimension.
Finally, we consider case (c). Consider three points p 1 , p 2 , p 3 ∈ Z lying on a line T ⊂ PW * . Then T must be an irreducible component of each conic through Z. Since Z spans the projective plane PW * , there must also be a point p 0 ∈ Z outside of T . All points in Z \ {p 0 } must lie on the line T , as otherwise there could be at most one conic passing through Z. If Z := {p 0 , p 1 , p 2 , p 3 } ⊂ Z, then each conic passing through Z also passes through Z, i.e. I(Z) 2 = I(Z ) 2 .
We may choose a basis z 0 , z 1 , z 2 of W such that Z ⊂ PW * with respect to these coordinates is given by The plane L := V (x 1 + x 2 − x 3 ) ⊂ P 3 is the image of P 2 [z 0 :z 1 :z 2 :z 1 +z 2 ] −−−−−−−−→ P 3 , so Z can be viewed as the finite set of points associated to L . Lemma 4.1 shows that L •2 ⊂ P 3 coincides with the image of the morphism ψ : PW → P(Sym 2 W/I(Z ) 2 ). On the other hand, Lemma 4.1 shows that L •2 ⊂ P n is a linear re-embedding of PW → P(Sym 2 W/I(Z) 2 ). From I(Z) 2 = I(Z ) 2 , we deduce that L •2 ⊂ P n is a linear re-embedding of the quadratic surface

Squaring in high ambient dimensions
Consider the case of k-dimensional linear spaces in P n for n k. For a general linear space L ∈ Gr(k, P n ), the finite set of points Z does not lie on a quadric. We know from Proposition 4.2 that the coordinate-wise square L •2 is a linear re-embedding of the k-dimensional second Veronese variety.
In this subsection, we investigate the first degenerate case where Z lies on a unique quadric. In the proposition that follows we point out that this case is closely related to studying symmetric matrices with a degenerate spectrum of eigenvalues. Here, we interpret P Sym 2 k k+1 (for k = R or C) as the projective space consisting of symmetric (k + 1) × (k + 1)-matrices up to scaling with entries in k.
Proposition 4.8. Let X ⊂ P Sym 2 R k+1 be the set of real symmetric (k + 1) × (k + 1)matrices with an eigenvalue of multiplicity ≥ k. Its Zariski closure in P Sym 2 C k+1 is the projective cone over L •2 , where L ⊂ P n is any k-dimensional linear space whose point configuration Z lies on a unique and smooth quadric.
Proof. Let L ⊂ P n be a k-dimensional linear space such that I(Z) 2 is spanned by a smooth quadric q ∈ P Sym 2 W . Choosing coordinates of W ∼ = C k+1 , we identify points in P Sym 2 W with complex symmetric (k + 1) × (k + 1)-matrices up to scaling and we can assume q = id ∈ P Sym 2 W . The second Veronese variety ν 2 (PW ) ⊂ P Sym 2 W consists of rank 1 matrices. Let X 0 ⊂ P(Sym 2 W/ q ) be the image of ν 2 (PW ) under the natural projection. By Lemma 4.1, X 0 is the coordinate-wise square L •2 up to a linear re-embedding.
The projective cone over X 0 ∼ = L •2 is the subvariety X 1 ⊂ P Sym 2 W consisting of complex symmetric matrices M such that the set M + id contains a matrix of rank ≤ 1. We observe that the rank of M − λ id is the codimension of the eigenspace of M with respect to λ ∈ C. Hence, We are left to show that X 1 is the Zariski closure in P Sym 2 C k+1 of X ⊂ P Sym 2 R k+1 . Since real symmetric matrices are diagonalizable, the multiplicity of an eigenvalue is the dimension of the corresponding eigenspace. Hence, X 1 ∩ P Sym 2 R k+1 = X. The set X is the orbit of the line V := {diag(λ, . . . , λ, µ) | [λ : µ] ∈ P 1 R } under the action of O(k + 1). The action is given by conjugation with orthogonal matrices and the stabiliser is O(k)×{±1}. Therefore, X has real dimension dim V +dim O(k+1)−dim O(k) = k+1. Also, X 1 is the projective cone over X 0 ∼ = L •2 , so it is a (k + 1)-dimensional irreducible complex variety. We conclude that X 1 is the Zariski closure of X in P Sym 2 C k+1 .
In fact, for s ≥ 3, we show that the claim holds scheme-theoretically, see Remark 4.15. We believe that in fact for arbitrary s the claim is even true ideal-theoretically.
First, we observe that Theorem 4.9 follows directly from Proposition 4.10.
Proof of Theorem 4.9. Analogous to the proof of Proposition 4.8, we identify P Sym 2 W with P Sym 2 C k+1 such that q = I s . By Lemma 4.1, the coordinate-wise square L •2 is a linear re-embedding of the variety obtained by the projection of ν 2 (PW ) from the point q = I s ∈ P Sym 2 W . Note that V (J) describes the set of points Y ∈ P Sym 2 W lying on the line joining q with some point in ν 2 (PW ). Hence, the projection from q is given by intersecting V (J) with a hyperplane H ⊂ P Sym 2 W not containing q = I s . From Proposition 4.10, we know that V (J) ∩ H is set-theoretically cut out inside H ∼ = P ( k+2 2 )−2 by the indicated number of quadrics and cubics. The coordinate-wise square L •2 is by Lemma 4.1 obtained as the image of V (J) ∩ H under a linear embedding ϑ : H → P n , leading to additional n − k+2 2 + 2 linear forms vanishing on L •2 .
We prove Proposition 4.10 in several steps. First, we describe a set X of certain lowdegree polynomials in the ideal J. Secondly, we show that V (X ) = V (J). Finally, we identify a subset of X providing minimal generators of the ideal (X ) ⊂ C[y], consisting of the claimed number of quadratic and cubic forms.
Proof. Using that  Proof. For k ≤ 5, we have checked computationally with a straightforward implementation in Macaulay2 [GS] that even the ideal-theoretic equality (X ) = J holds. We now argue that from this we can conclude the claim for arbitrary k.
(i) Let s ≥ 3. We need to show that the ideal generated by X ⊂ C[y] coincides with J ⊂ C[y] after localisation at any element in the set since the union of the corresponding non-vanishing sets D(y ij ), D(y ii − y jj ) is U 1 . In order to show that (X ) and J agree after localisation at y i 0 j 0 for {i 0 } ∩ {j 0 } ⊂ {s + 1, . . . , k + 1}, we may substitute y i 0 j 0 = 1 in both ideals. For a fixed 0 ≤ s distinct from i 0 and j 0 , we note that t+Y i 0 0 |j 0 0 = M i 0 0 |j 0 0 ∈ J 0 | y i 0 j 0 =1 . Hence, eliminating t from J 0 | y i 0 j 0 =1 just amounts to replacing t = −Y i 0 0 |j 0 0 in each occurrence of t in the minors M ij| m (for i = j, = m) generating the ideal J 0 .
According to (4.1), this leads to the following generators of J| y i 0 j 0 =1 : , we need to check that each of these polynomials belong to (X )| y i 0 j 0 =1 . For this, it is enough to see that they can be expressed in terms of those polynomials in X that only involve variables with indices among {i 0 , j 0 , 0 , i, j, }. This corresponds to showing the claim for a corresponding symmetric submatrix of M of size at most 6 × 6. We conclude that it is enough to check Similarly, in order to show that J| y i 0 i 0 −y j 0 j 0 =1 = (X )| y i 0 i 0 −y j 0 j 0 =1 holds for i 0 , j 0 ≤ s distinct, we realise that t+Y i 0 0 |i 0 0 −Y j 0 0 |j 0 0 = M i 0 0 |i 0 0 −M j 0 0 |j 0 0 ∈ J 0 | y i 0 j 0 =1 holds for fixed 0 ≤ s distinct from i 0 and j 0 . Therefore, replacing t = Y j 0 0 |j 0 0 − Y i 0 0 |i 0 0 in the expressions for the 2 × 2-minors of M describes generators of J| y i 0 i 0 −y j 0 j 0 =1 . As before, these polynomials involve variables with at most six distinct indices, so it is enough to verify the claim for k ≤ 5 by the same argument as above.
(ii) For s = 2, the argument from (i) still shows J 0 | y i 0 j 0 =1 = (X )| y i 0 j 0 =1 for {i 0 , j 0 } ∩ {3, . . . , k + 1} = ∅. For the localisation at y 12 and at y 11 − y 22 , the argument does not apply since we cannot choose 0 distinct from {i 0 , j 0 } = {1, 2} as before. Hence, we have shown the equality of V (X ) and V (J) only on U 2 . (iii) We observe that the polynomials in X vanish on the point I s ∈ P Sym 2 C k+1 , and that I s ∈ V (J) by definition of J. Together with (i), this proves the claim for s ≥ 3. For s = 2, the polynomials in X vanish on all symmetric matrices of the form A = a c c b 0 0 0 ∈ Sym 2 C k+1 . On the other hand, each such matrix is a point in V (J), since A+t 0 I 2 is a matrix of rank ≤ 1 for t 0 ∈ C such that t 2 0 +(a+b)t 0 +(ab−c 2 ) = 0. Together with (ii), we conclude that V (X ) = V (J) holds set-theoretically.
Lemma 4.13. The vector spaces spanned by the polynomials in X satisfy: so H 1 and H 2 lie in the ideal generated by E, F and G.
Next, we identify maximal linearly independent subsets of E, F, G.
Lemma 4.14. The following sets are bases for the vector spaces E , F and G : . . , s}}, Proof. The polynomials in E not contained in B E ∪ (−B E ) are the polynomials Y ij| m for i < j < < m. However, these can be expressed as so B F spans F . Each of the polynomials Y i |im − Y j |jm in B F contains a monomial not occurring in any of the other polynomials of B F , namely y ii y m . Therefore, the polynomials in B F are linearly independent. For 3 ≤ m ≤ s − 1 and ∈ {3, . . . , m − 1} ∪ {s}, the polynomial Y 12|12 − Y 2 |2 + Y m| m − Y m1|m1 is the unique polynomial in B G containing the monomial y y mm . In particular, if a linear combination of polynomials in B G is zero, none of the above polynomials can occur in this linear combination. The remaining polynomials in B G are of the form Y 1s|1s − Y s2|s2 + Y 2m|2m − Y m1|m1 for 3 ≤ m ≤ s − 1. Among these, the polynomial containing the monomial y 22 y mm is unique. We conclude that the polynomials in B G are linearly independent.
The vector space of symmetric s × s-matrices with zero diagonal and whose columns all sum to zero is of dimension s 2 − s, so dim C G ≤ s 2 − s. On the other hand, we can count that |B G | = s−3 2 + 2(s − 3) = s 2 − s, so B G is a basis of G . Proof of Proposition 4.10. By Lemma 4.12, V (J) = V (X ) holds set-theoretically. For s = 3, we observe that H 1 ∪ H 2 consists up to sign of seven linearly independent cubics, so by Lemma 4.13, the ideal (X ) is in this case minimally generated by those seven cubics and the polynomials in B E , B F and B G from Lemma 4.14.
Remark 4.15. In fact, for s ≥ 3, our proof shows that V (X ) is the same scheme as V (J) away from the point I s ∈ P Sym 2 C k+1 . In the proof of Theorem 4.9, we considered V (J) ∩ H, where H is a hyperplane not containing I s . Since V (J) ∩ H = V (X ) ∩ H scheme-theoretically, we conclude that our equations for L •2 in Theorem 4.9 cut out not only the correct set, but even the correct scheme. In fact, we believe that we have ideal-theoretic equality for the specified set of polynomials, but our proof stops short of verifying this.
In the proof of Proposition 4.10, we have shown the following result about eigenspaces of symmetric matrices.
Corollary 4.16. Let s ≥ 4. A symmetric matrix A ∈ C s×s has an eigenspace of codimension ≤ 1 if and only if its 2 × 2-minors satisfy the following for i, j, k, ≤ s distinct: These equations describe the Zariski closure in the complex vector space Sym 2 C s of the set of real symmetric matrices with an eigenvalue of multiplicity ≥ s − 1.
Proof. A complex symmetric matrix A ∈ C s×s has an eigenspace of codimension 1 with respect to an eigenvalue λ ∈ C if and only if the matrix A − λ id is of rank 1, which means that A ∈ V (J) for the case s = k + 1. By Lemma 4.12, this is equivalent to the vanishing of the equations X , which are the above relations among 2 × 2-minors for s = k + 1 ≥ 4. The second claim has been proved in Proposition 4.8.
The proof of Theorem 4.9 was based on relating the coordinate-wise square L •2 in the case dim C I(Z) 2 = 1 to the question when a symmetric matrix can by completed to a rank 1 matrix by adding a multiple of I s . In the same spirit, for arbitrary linear spaces L (no restrictions on the set of quadrics containing Z), determining the ideal of the coordinate-wise square L •2 boils down to the following problem in symmetric rank 1 matrix completion: Indeed, let L be an arbitrary linear space of dimension k and let B ∈ C (n+1)×(k+1) be a chosen matrix of full rank describing L as the image of the linear embedding P k → P n given by B. Then the rows of B form the finite set of points Z ⊂ (P k ) * . Identifying quadratic forms on P k with symmetric (k + 1) × (k + 1)-matrices, the subspace I(Z) 2 ⊂ Sym 2 (C k+1 ) * corresponds to I(Z) 2 = {P ∈ C (k+1)×(k+1) symmetric such that BP B T has a zero diagonal}.
By Lemma 4.1, the coordinate-wise square L •2 is a linear re-embedding of projecting the second Veronese variety ν 2 (P k ) = {rank 1 symmetric (k + 1) × (k + 1)-matrices up to scaling} from P(I(Z) 2 ), so describing the ideal of L •2 corresponds to solving Problem 4.17 for the given matrix B. Similarly, describing the coordinate-wise r-th power of a linear space corresponds to the analogous problem in symmetric rank 1 tensor completion.
By Lemma 4.1, determining the coordinate-wise r-th power of a linear space corresponds to describing the projection of the r-th Veronese variety from a linear space of the form P(I(Z) r ) for a non-degenerate finite set of points Z. We may ask how general this problem is, and pose the question which linear subspaces of P Sym r W are of the form P(I(Z) r ): Question 4.18. Which linear subspaces of C[z 0 , . . . , z k ] r can be realised as the set of degree r polynomials vanishing on some non-degenerate finite set of points in P k of cardinality ≤ n + 1?
We envision that an answer to this question may lead to insights into describing which varieties can occur as the coordinate-wise r-th power of some linear space in P n .