Asymptotics of degrees and ED degrees of Segre products

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product $X\times Q_{n}$, where $X$ is a projective variety and $Q_n\subset \mathbb{P}^{n+1}$ is a smooth quadric hypersurface.


Introduction
Let V R be a real vector space equipped with a distance function and let X ⊂ P(V R ⊗ R C) be a complex projective variety. Two fundamental features of X are its degree and its Euclidean Distance degree (ED degree). While the first is one of the basic numerical invariants of an algebraic variety, the second was recently introduced in [DHOST] and since then has found several interesting applications in Pure and Applied Algebraic Geometry [DH, QCL, DLOT, DOT, BKL, HL, HW].
The aim of this paper is to initiate a study of the asymptotic behavior of these two important notions attached to some special varieties and their duals. These varieties are Segre products, i.e. images of direct products of projective varieties through the Segre embedding. Our perspective is naturally inspired by the recently emerging interest in stabilization properties in Algebraic Geometry and Representation Theory, seeking for results about large families of related varieties at once, rather than specific instances. The discussion around [DH,Conjecture 1.3] motivated this work. Indeed we found the geometrical explanation sought in [DH] (see Corollary 4.14 and its proof). We hope that the specialization technique of this paper may open the road towards new results, including the above conjecture.
Whenever a classical Segre variety P k 1 × · · · × P k d ⊂ P(C k 1 +1 ⊗ · · · ⊗ C k d +1 ) is not dual defective, i.e. its dual variety is a hypersurface, the polynomial defining the latter is the hyperdeterminant [GKZ,Chapter 14]. These higher analogues of matrix determinants are of utmost importance and yet their properties are far from being completely understood. Hyperdeterminants play a prominent role behind all the results in this paper.
We now showcase our results. We start off from the very classical (hypercubical) hyperdeterminants of the Segre variety P(C n+1 ) ×d establishing an asymptotic formula of its degree as the dimension n + 1 of the vector space goes to infinity.
Theorem (Theorem 3.8). Let N (n1 d ) be the degree of the hyperdeterminant of format (n + 1) ×d . Then asymptotically, for d ≥ 3 and n → +∞, This should be compared to the behavior of the ED degree (with respect to the Frobenius inner product introduced in Definition 4.1) of the Segre variety X = P(C n+1 ) ×d for d ≥ 3 and n → +∞, found by Pantone [Pan]: The paper is organized as follows. In §2, we fix notation and introduce the terminology used throughout the article. In §3, we derive the asymptotic formula for the degree of the (hypercubical) hyperdeterminants. As an additional result, in §3.2 we give an asymptotic formula for the degree of the hyperdeterminant of format 2 ×d as d goes to infinity. In §4, we first provide a description of critical points for tensors of boundary format as in Theorem 4.12. Thereafter we show Theorem 4.13 and Corollary 4.14, along with recording other observations. In §5, we recall polar classes and utilize them to establish the stabilization of the degree featured in Corollary 5.6. In §5.1, we mention some by-products of the stabilization above and formulate some intriguing conjectures, naturally emerging from our experiments.
Definition 2.1. A tensor t ∈ V is of rank one (or decomposable) if t = v 1 ⊗ · · · ⊗ v d for some vectors v j ∈ V j for all j ∈ [d] := {1, . . . , d}. Tensors of rank at most one in V form the affine cone over the Segre variety of format (n 1 + 1) × · · · × (n d + 1), that is the image of the projective morphism for all non-zero v j ∈ V j . The Segre variety introduced above is often denoted simply by P(V 1 ) × · · · × P(V d ).
For the ease of notation, in the rest we will abuse notation identifying a tensor t ∈ V with its class in projective space.
Definition 2.2 (Dual varieties). Let X ⊂ P(W ) be a projective variety, where dim(W ) = n + 1. Its dual variety X ∨ ⊂ P(W * ) is the closure of all hyperplanes tangent to X at some smooth point [GKZ,Chapter 1]. The dual defect of X is the natural number δ X := n − 1 − dim(X ∨ ). A variety X is said to be dual defective if δ X > 0. Otherwise, it is dual non-defective. When X = P(W ), taken with its tautological embedding into itself, X ∨ = ∅ and codim(X ∨ ) = n + 1.
Of particular interest are dual varieties of Segre varieties, whose non-defectiveness is characterized by the following result.

Asymptotics of degrees of some hyperdeterminants
In this section, we establish the asymptotic results for hyperdeterminants of formats (n + 1) d (as n → +∞) and formats 2 d (as d → +∞).

3.1.
Asymptotics for the hyperdeterminant of (P n ) ×d . Let x = (x 1 , . . . , x d ) be a coordinate system in C d . Let α = (α 1 , . . . , α d ) ∈ N d be a d-tuple of natural numbers. Define x α := x α 1 1 · · · x α d d . Raichev and Wilson [RW] gave a method to find the asymptotic behavior of the coefficients of a multivariate power series α∈N d f α x α in the variables x 1 , . . . , x d , which is the Taylor expansion of a function F = G/H p , where G and H are holomorphic functions in a neighborhood of the origin of C d . They showed that the asymptotics of f nα , for α ∈ N d , is governed by special smooth points of the complex (analytic) variety V := {H(x) = 0} ⊂ C d .
Proof. The first part follows immediately from the definition of H. For the second part, we have where we use the convention a b = 0 if b < 0. The matrix M just defined has equal rows. We found this construction convenient for visualizing the sums. By a direct computation, one verifies that the sum of the entries strictly below the diagonal of M is On the other hand, the sum of the entries strictly below the diagonal of M is , strictly minimal, critical, isolated, and non-degenerate.
Proof. We first show that the point c This choice of c is the one made by Pantone in [Pan]. We have namely c belongs to V. Moreover, by Lemma 3.5 we have that It remains to show that the point c is non-degenerate. By [RW,Definition 3.1], the point c is non-degenerate whenever the quantity detg ′′ (0) = 0. By [RW,Proposition 4.2], By Lemma 3.5, we have the identities Substituting these values in the expressions for q and detg ′′ (0), we observe that c is non-degenerate: Proposition 3.7. The following identity holds true: Proof. The quantity L k (ũ j ,g) is defined in the statement of [RW,Theorem 3.2]. For k = 0, we have L 0 (ũ j ,g) =ũ j (c). By [RW,Proposition 4.3], we haveũ 0 (c) = ) 2 , which gives the second equality in the statement.
Theorem 3.8. Let N (n1 d ) be the degree of the hyperdeterminant of format (n + 1) ×d . Then asymptotically, for d ≥ 3 and n → +∞, Proof. By Theorems 3.3 and 3.4, for N = 1 and c = ( 1 d−1 , . . . , 1 d−1 ), we have Since the second and third summand in the square brackets are both O(n Conclusion follows by plugging in the identity for L 0 (ũ 0 ,g) of Proposition 3.7.
Remark 3.9. For d = 3 and d = 4 we recover the approximations The approximation formula for d = 3 appears also in [Slo, A176097] and it is interesting to compare it with the asymptotic result proved by Ekhad and Zeilberger for the EDdegree which is (see [EZ, Pan]) Remark 3.10 (A Segre-Veronese hyperdeterminant). Theorem 3.8 can be generalized further to a special class of Segre-Veronese varieties, with essentially the same calculations. We preferred to keep the statement for hypercubical hyperdeterminants of Segre varieties for the ease of notation.
Segre-Veronese varieties are obtained combining the Veronese embedding v ω already defined with the Segre embedding Seg introduced in Definition 2.1. More precisely, let (ω 1 , . . . , ω d ) ∈ N d . The degree (ω 1 , . . . , ω d ) Segre-Veronese embedding of P(V 1 ) × · · · × P(V d ) is the Segre embedding of the product v ω 1 P(V 1 ) × · · · × v ω d P(V d ), which we keep calling v ω 1 P(V 1 ) × · · · × v ω d P(V d ) for simplicity. Now let N (k 1 , . . . , k d ; ω 1 , . . . , ω d ) be the degree of the hyperdeterminant, namely the polynomial defining the dual hypersuperface, when defined, of v ω 1 P(V 1 ) × · · · × v ω d P(V d ). Assume that ω 1 = · · · = ω d = ω for some ω ∈ N. Applying [GKZ,Theorem 2.4], one verifies that the generating function of the degrees N d (k 1 , . . . , k d ; ω d ) is given by In order to apply again Theorem 3.3, a possible choice of a point is c = 1 ωd−1 , . . . , 1 ωd−1 . Then asymptotically, for d ≥ 3 and n → +∞ Remark 3.11 (The case of the discriminant). Now we focus on a special case of Remark 3.10, namely when d = 1. Here, the number N (n; ω) is the degree of the discriminant polynomial ∆ n,ω , i.e., the degree of the dual variety of the Veronese variety X = v ω P(V ). It is interesting to recall that when ω is even, denoting by x 2 the squared Euclidean norm of x, for any f ∈ S ω V the polynomial where x ω = x 2 ω/2 , vanishes exactly at the eigenvalues of f (see [QZ,Theorem 2.23], [Sod18,Theorem 3.8]). This is a beautiful result first discovered by Liqun Qi, who called ψ f the E-characteristic polynomial of f . Its degree is equal to where the quantity EDdegree F (X) is recalled in Definition 4.2. The identity (3.3) was proved in [CS,Theorem 5.5] and is a particular case of [FO,Theorem 12]. However, this result had already essentially been known in complex dynamics by the work of Fornaess and Sibony [FS].
From equation (3.3), we see that EDdegree F (X) is much less than the degree of ∆ n,ω , which is N (n; ω) = (n + 1)(ω − 1) n . The reason of this huge degree drop in (3.2) is that the polynomial x ω (for ω > 2) defines a non-reduced hypersurface, hence highly singular, where ∆ n,ω vanishes with high multiplicity: having a root of high multiplicity in (3.2) for λ = +∞ corresponds to a large degree drop. Our asymptotic analysis for the hyperdeterminant originally arose from the desire to understand whether some analogous result could have been true in the non-symmetric setting. One has the ratio One might also analyze the asymptotics of N (n; ω) and EDdegree F (X) for n fixed and ω → +∞. In this case, we have the ratio Instead of EDdegree F (X), we might consider the generic ED degree of X, recalled in Definition 4.3. It was shown in [DHOST,Proposition 7.10] that .21 for a more general case). From the last formula, we obtain the ratio N (n; ω) EDdegree gen (X) ≈ n + 1 2 n+1 − 1 as ω → +∞ .

3.2.
Asymptotics for the hyperdeterminant of (P 1 ) ×d . The degree of the hyperdeterminant of X = (P 1 ) ×d is denoted by N (1 d ).
Here we analyze the asymptotics with respect to d. By [GKZ,Corollary 2.10,Chapter 14], the exponential generating function for Proof. We have Therefore, Both the inner sums converge to e −2 as d → +∞. Using the Stirling approximation for the factorial we obtain the desired asymptotic formula.
Remark 3.13. Applying Theorem 4.6, one verifies that Moreover, the generic ED degree of X is (see Remark 4.21 for a more general formula) The two inner sums converge to e −1 and e −2 as d → +∞, respectively. Using (3.4) we obtain the asymptotic formulas as d → +∞ .
In particular, the degree of the hyperdeterminant N (1 d ) grows faster than EDdegree F (X) and slower than EDdegree gen (X) as d → +∞.

Stabilization of the ED degree of some Segre varieties
Throughout the section, we let V R 1 , . . . , V R d be real vector spaces of dimensions dim(V R i ) = n i + 1, respectively. Recall from §2 that a real tensor of format (n 1 + 1) × · · · × (n d + 1) is a multilinear map t : , an element of the tensor product (over R) j comes equipped with an inner product (and so with a natural distance function). Their tensor product V R inherits a natural inner product defined as follows.
Definition 4.1. The Frobenius inner product of two real decomposable tensors t = x 1 ⊗ · · · ⊗ x d and and it is naturally extended to every vector in V R . When V R j is equipped with the standard Euclidean inner product for all j ∈ [d], one finds that Analogously to what happens in a Euclidean space, it is natural to look at critical points of the distance function δ F (t, ·) : X R → R from a given tensor t ∈ V R with respect to some special sets X R ⊂ V R . The most relevant for our purposes is the real affine cone over the Segre variety . This leads us to the more general definition of ED degree of an affine variety to be discussed in a moment.
Let V R be a real vector space equipped with a distance function δ : be a real affine variety and let u ∈ V be general. Consider the complex vector space V := V R ⊗ C and the complex variety X := X C . The distance function δ is extended to a complex-valued function δ : V × V → C (which is not a Hermitian inner product). The point is that even though the function δ is truly a distance function only over the reals, the complex critical points of δ on X are important to draw all the metric information about the real affine cone X R .
Definition 4.2 (ED degree [DHOST]). The Euclidean distance degree (ED degree) of X is the (finite) number of complex critical points of the function δ(u, _) : X \ X sing → C, where X sing is the singular locus of X. We denote it by EDdegree δ (X) in order to stress its dependence on δ.
Definition 4.3 (Generic ED degree). The isotropic quadric associated to δ is the quadric hypersurface . When X is transversal to Q δ (this assumption holds for a general Q δ ), the ED degree of X with respect to δ is called generic ED degree of X and it is denoted by EDdegree gen (X).
In the following, we focus on the case when Given a tensor t ∈ V , we refer to (complex) critical points of δ F (t, ·) on X simply as critical points. Lim [L] and Qi [Q] independently defined singular vector tuples of tensors and associated them to non-isotropic critical points of the distance function from the affine cone over the Segre variety X R . The next result is a reformulation of [L,Eq. (9)] and of [FO,Lemma 19].
Theorem 4.4. Given a real tensor t ∈ V R , the non-isotropic decomposable critical points of t correspond to tensors for some σ ∈ C, called a singular value of t corresponding to v. The corresponding d-ple x (1) , . . . , x (d) is called singular vector d-ple of t. (In view of these results, we shall refer to singular vector tuples simply as critical points.) For each j ∈ [d], equation (4.3) may be written as where, on the left-hand side, we have a contraction of the tensor t along the j-th direction. In analogy with matrices, the best rank one approximation problem for t is solved as indicated in the following result by Lim [L,Eq. (9)] which we reformulate similarly to [FO,Theorem 20].
Theorem 4.5. Let t ∈ V R be a real tensor. Then t admits real singular values and real critical points. Suppose σ is a real singular value of t such that σ 2 is maximum, and assume v = σ x (1) ⊗ · · · ⊗ x (d) is a critical point corresponding to σ. Then v is a best rank one approximation of the tensor t. Moreover, a best rank one approximation of t is unique if t ∈ V is general.
The number of singular vector d-ples of a general tensor t ∈ V , i.e., the ED degree of the Segre variety X = P(V 1 ) × · · · × P(V d ) with respect to the distance function δ F in V R , is the content of the next result.
Now assume that the tensor t ∈ V is expressed in coordinates by the multidimensional array Then the critical points of t are of the form σ(x (1) ⊗ · · · ⊗ x (d) ) ∈ V , with no zero component, and satisfy equations (4.4) which can be rewritten as Eliminating the parameter σ ∈ C, one derives the multilinear relations (for all 1 ≤ k < s ≤ n j + 1 and for all j ∈ [d]) that all critical points must satisfy: Definition 4.7 (Critical space of a tensor). The critical space (or singular space) H t of the tensor t ∈ V is the linear projective space defined the equations (in the unknowns z i 1 ···i d that serve as linear functions on V ) Remark 4.8. The tensor t belongs to its critical space H t [OP,§5.2]. For a general tensor t, let Z t denote the set of critical points and consider its projective span Z t . Then Z t ⊂ H t and [DOT,§3.5] shows that they coincide for formats (n 1 + 1) × · · · × (n d + 1) satisfying a triangle inequality (i.e. the so-called sub-boundary format). However, they do not concide in every format. For the Segre variety of format 2 × 2 × 4, Z t consists of 8 critical points with Z t ∼ = P 6 , but H t ∼ = P 7 .
Lemma 4.9. Let X = P(V 1 ) × · · · × P(V d ) × P(W ) with dim(W ) = m + 1. Let t ∈ V ⊗ W be a non-concise tensor, i.e. there exists a proper subspace L ⊂ W such that t ∈ V ⊗ L. Then either a critical point is in X ∩ P(V ⊗ L) or its singular value is zero.
Proof. Fix bases for the vector spaces V i and a basis {y i } for W , and assume dim(L) = ℓ + 1 ≤ m + 1.
By assumption, there exists a change of bases such that we may write the tensor t ∈ V ⊗ W as Now, let v = x 1 ⊗ · · · ⊗ x d ⊗ z ∈ X be a critical point of t with a non-zero singular value σ = 0. By their defining equations (4.4), we have Denote by {y * j } ⊂ W * the dual basis of {y j }. Since t ∈ V ⊗ L, and since z is the result of a contraction of t, this vector satisfies y * j (z) = 0 for all ℓ + 1 ≤ j ≤ m. However, in the given basis, these are the defining equations of P(L) ⊂ P(W ) and so v ∈ X ∩ P(V ⊗ L).
Remark 4.10. Let d = 3 and n 1 = n 2 = m = 1. A general tensor t ∈ V ⊗ W has 6 distinct singular values. Assume that t is non-concise, in particular t ∈ V ⊗ L, where L ⊂ W with dim(L) = 1. We consider the ED polynomial EDpoly X ∨ ,t (ε 2 ) of the dual variety of X = P(V 1 ) × P(V 2 ) × P(W ) at t (see §5.1 for the definition of ED polynomial). It turns out that the roots of EDpoly X ∨ ,t (ε 2 ) are the squared singular values of t (see [Sod20,Proposition 5.1.4]). The second author computed symbolically in [Sod20,§5.4] the ED polynomial EDpoly X ∨ ,u (ε 2 ), for any u ∈ V ⊗ W , as a univariate polynomial in ε 2 of degree 6 whose coefficients are homogeneous polynomials in the entries u ijk of u. In particular, when u = t and assuming that V ⊗ L has equations t 112 = t 122 = t 212 = t 222 = 0, one verifies by direct computation that i.e., the variety of 2 × 2 matrices of rank one. In particular, two of the singular values of t are non-zero and correspond to critical points in Y . On the other hand, the factor ε 8 = (ε 2 ) 4 tells us that the remaining 4 singular values of t are zero, thus confirming Lemma 4.9.
Recall that the Segre variety P n 1 × · · · × P n d × P N , where N = d i=1 n i , and the corresponding hyperdeterminant are said to be of boundary format. This format turns out to be important for our purposes.
Keeping the notation from above, let dim(V i ) = n i + 1 and dim(W ) = m + 1. Let t ∈ V ⊗ L ⊂ V ⊗ W , where L ⊂ W is a hyperplane, namely t is non-concise and has the last slice zero. Then the hyperdeterminant vanishes on t, i.e. Det(t) = 0.
By definition, this means that the tensor t is degenerate: there exists a non-zero decomposable tensor (4.6) The kernel K(t) of a tensor t is the variety of all non-zero v 1 ⊗ · · · ⊗ v d ⊗ z ∈ V ⊗ W such that (4.6) is satisfied. The description of the critical points of t outside the hyperplane L can be given in terms of K(t), this is the content of our next result. has rank N , i.e., t ∈ V ⊗ L where L ⊂ V is a subspace of dimension N . Assume that t is general with this property. Then (i) the kernel K(t) of t consists of N ! i n i ! linear spaces of projective dimension m − N corresponding to the intersection of the kernel of the flattening map (4.7) with the Segre variety of rank one matrices X = P(V 1 ) × · · · × P(V d ), which has degree N ! i n i ! , (ii) the points of K(t) are exactly the critical points of t with zero singular value. Moreover, the latter critical points of t are the only ones not lying on V ⊗ L. In fact, they lie on its orthogonal complement V ⊗ L ⊥ . Proof.
(i) The projectivization P(Ker(π W )) of the kernel of the flattening map (4.7) has codimension N in P(V ). By the genericity assumption on t, the intersection P(Ker(π)) ∩ X is given by deg(X) = N ! i n i ! points. (ii) By Lemma 4.9, each critical point of t is either in V ⊗ L or it has zero singular value. On the other hand, since the tensor t is general in V ⊗ L, we may assume Det(t) = 0. This implies that every critical point of t in V ⊗ L has a non-zero singular value. As a consequence, the only critical points outside V ⊗ L are the ones with singular value zero. To see where they are located and thus establishing the last sentence, we proceed as follows. Since our vector spaces V and W are equipped with an inner product, they come with an indentification with their duals: V ∼ = V * and W ∼ = W * . Therefore the flattening map π W above may be regarded as a linear map π W : V → W . Note that the other flattening map π V : W → V induced by t is dual to π W . In bases, this amounts to say that π V = π T W . Now, suppose that t has a critical point v ⊗ z ∈ V ⊗ W with singular value σ, where v = v 1 ⊗ · · · ⊗ v d is a decomposable tensor and z ∈ W . By definition, this means that v and z are non-zero with π W (v) = σz and π T W (z) = σv. Assume z / ∈ L. As noticed above, this critical point has singular value σ = 0. Since t ∈ V ⊗ L, we have Im(π W ) = L. Note that Ker(π T W ) = Im(π W ) ⊥ = L ⊥ . Since π T W (z) = σv = 0, one finds z ∈ Ker(π T W ) = L ⊥ . In conclusion, the critical points of a general tensor t, that are outside V ⊗ L, lie on its orthogonal complement.
Using directly equations (4.4) satisfied by the critical points, we also show the following result.
Proof. We show that the critical space of t in P(V ⊗ W ) lies inside P(V ⊗ C N +1 ). We may assume dim(W ) = N + 2.
Consider the equations (4.4) for j = d, s = N + 2. Since by assumption t i 1 ···i d ,N +2 = 0, these relations simplify and we obtain: The equations inside the brackets have no non-zero solutions by the assumption Det(t) = 0 and by the description in [GKZ,Chapter 14,Theorem 3.1]. Hence x (d+1) N +2 = 0, which proves the statement.
Corollary 4.14. Let N = d i=1 n i . For all m ≥ N , we have EDdegree F (P n 1 × · · · × P n d × P m ) = EDdegree F P n 1 × · · · × P n d × P N . (4.9) Proof. Note that, given a smooth projective variety X ⊂ P(V ), when the number of critical points (of the distance function with respect to X) of a given point t ∈ V is finite, it coincides with EDdegree(X).
In other words, we may consider convenient specializations in order to compute the ED degree. To evaluate the ED degree on the left-hand side of (4.9), we specialize t as in Theorem 4.13. Then the result of Theorem 4.13 shows that the critical points of t are the same as the ones needed to compute the right-hand side of (4.9).
Theorem 4.13 generalizes to the partially symmetric case. The proof is analogous, following the critical space in the partially symmetric case, as defined in [DOT].
Remark 4.17. Corollary 4.16 does not hold if the last factor P m is replaced by v s P m for some integer s ≥ 2. This can be checked also from the formula in [FO,Theorem 12] (generalizing Theorem 4.6 to the case of partially symmetric tensors) which does not stabilize anymore for m → +∞.
Example 4.19. As an illustration of Conjecture 4.18, we consider the Segre products X ×P m for m ≥ 0 and for some choices of X. The values of EDdegree F (X × P m ) are listed in Table 2 for the different varieties X. The entries of the i-th column in Table 3 correspond to the values of EDdegree F (X ∩H i−1 ).
The numbers in Table 2 as well as in the first column of Table 3 are computed according to Theorem 4.6. The boxed ED degrees in Table 3 have been checked numerically with the software Julia [BT], and the remaining ones with the software Macaulay2 [GS].
Observe that each number in the i-th column of Table 2 is the sum of the first i entries in the corresponding row of Table 3, thus confirming Conjecture 4.18.  Remark 4.20. Using the notations of Theorem 4.12, we assume d = 2. Let t ∈ V ⊗L, where L ⊂ W is a linear subspace. The critical points of t generally fill up several components of different dimensions, forming the critical locus. These components are either in V ⊗ L or in V ⊗ L ⊥ . By the description of the critical points in Theorem 4.12, the critical locus of a general t sitting inside V ⊗ L ⊥ coincides with the contact locus of t, see [Ott,§3] (these last two observations apply to all formats with any number of factors). For a general tensor t ∈ V ⊗ L, define C L and C L ⊥ to be the critical loci inside V ⊗ L and V ⊗ L ⊥ , respectively. In Table 4 we collect the dimensions and the degrees of these loci for the first few cases of boundary formats and where the linear subspace L is varying.

Stabilization of the degree of the dual of a special Segre product
In this section, we start introducing classical material on dual varieties. We refer to [GKZ] for details on the rich theory of projective duality. We shall demonstrate Theorem 5.4, showing a stabilization property of dual varieties to some Segre products.
Definition 5.1 (Segre products). Let X 1 ⊂ P(V 1 ) and X 2 ⊂ P(V 2 ) be two projective varieties. Their direct product X 1 × X 2 may be embedded in P(V 1 ⊗ V 2 ) via the Segre embedding introduced in Definition 2.1. The image of this embedding is called the Segre product of X 1 and X 2 and it is again denoted by X 1 × X 2 .
Definition 5.2. Let Y ⊂ P(V ) ∼ = P n be an irreducible projective variety of dimension m, where V is a Euclidean space. The Euclidean structure of V allows us to naturally identify V with its dual V * . The conormal variety of Y is the incidence correspondence where N z 1 Y denotes the normal space of Y at the smooth point z 1 .
A fundamental feature of the conormal variety is the content of the biduality theorem [GKZ, Chapter 1]: one has N Y = N Y ∨ . The latter implies (Y ∨ ) ∨ = Y , the so-called biduality.
The polar classes of Y are defined to be the coefficients δ i (Y ) of the class in cohomology , the maps π 1 , π 2 are the projections onto the factors of P(V ) × P(V ) and H, H ′ are hyperplanes in P(V ). If we assume Y smooth, δ i (Y ) may be computed utilizing the Chern classes of Y . These are the Chern classes of the tangent bundle T Y of Y . One computes [Hol,§3]: The right-hand side of (5.3) is always a nonnegative integer. The integer codim(Y ∨ ) − 1 equals the minimum i such that δ i (Y ) = 0. Whenever Y ∨ is a hypersurface, one has When Y is not smooth, we can replace Chern classes with Chern-Mather classes. They are constructed as follows. Let Y ⊂ P(V ) ∼ = P n be a projective variety of dimension m. We denote by G(m + 1, V ) the Grassmannian of (m + 1)-dimensional vector subspaces of V . Consider the Gauss map where Q denotes the quotient bundle. From this it follows that Q ⊗ U ∨ is isomorphic to the tangent bundle T G(m+1,V ) . The push-forwards under ν of the Chern classes of the universal bundle restricted to the Nash blow-up Y are the Chern-Mather classes c M i (Y ) of Y . They agree with Chern classes whenever Y is smooth.
The polar classes δ i (Y ) may be written in terms of the Chern-Mather classes c M i (Y ), thus generalizing the classical formula in (5.1). This generalization is due to Piene ([Pie88,Theorem 3] and [Pie78]), see also [Alu,Proposition 3.13]: In equation (5.3) we use a slightly different convention than in [Alu]. Indeed, for us c M i (Y ) is the component of dimension m − i (as with standard Chern classes), while in Aluffi's paper it is the component of dimension i. We have also the following generalization of equation (5.2), when Y ∨ is a hypersurface: Lemma 5.3. Let X be a projective variety of dimension m. For every integer n ≥ 0, let Y n ⊂ P n+1 be a smooth hypersurface of degree d. Then where for all i ∈ {0, . . . , m} 1+d y for the Chern-Mather polynomials of X and Y n , respectively. The expression for c(Y n ) is derived from the short exact sequence of sheaves 0 → T Yn → T P n+1 |Yn → N Yn/P n+1 → 0 , and applying Whitney formula. Keeping into account the relations x m+1 = 0 = y n+1 , we have where p s (X) is a homogeneous polynomial of degree s in the variables x and y. Using (5.4), the polar class δ 0 (X × Y n ) is given by (−1) s (m + n + 1 − s)p s (X) · (x + y) m+n−s .
A computation reveals that p s (X) · (x + y) m+n−s = a s (X) x m y n , where Plugging in the relations above in the definition of δ 0 (X × Y n ) and factoring out the Chern-Mather classes c M i (X) we derive that Therefore, for all i ∈ {0, . . . , m} the coefficient of deg(c M i (X)) is Theorem 5.4. Let X be a projective variety of dimension m. For every integer n ≥ 0, let Y n ⊂ P n+1 be a smooth hypersurface of degree d. Then Proof. For all i ∈ {0, . . . , m}, let α i = α i (n, m, d) be the coefficient of deg(c M i (X)) introduced in (5.5). In what follows, we use that a b = 0 if b is a negative integer and we shall sometimes use the formalism of gamma functions [SS,§6] for convenience.
A result due to Weyman and Zelevinsky detects the dual defectiveness of Segre products [WZ,Theorem 0.1] assumed in Corollary 5.6.
Theorem 5.5 (Weyman-Zelevinsky). Let X 1 and X 2 be (embedded) irreducible projective varieties. The dual variety (X 1 × X 2 ) ∨ is a hypersurface if and only if We are finally ready to state the main result of this section.
Corollary 5.6. Let X be a projective variety of dimension m. For n ≥ 0, let Q n ⊂ P n+1 be a smooth quadric hypersurface. Suppose that (X × Q m ) ∨ is a hypersurface. Then (X × Q n ) ∨ is a hypersurface of the same degree as (X × Q m ) ∨ for all n ≥ m.
Proof. By Theorem 5.5, the variety (X × Q m ) ∨ is a hypersurface if and only if m ≥ codim(X ∨ ) − 1.
If this condition is satisfied, then it is satisfied for all n ≥ m. In addition, equation (5.2) gives deg[(X × Q n ) ∨ ] = δ 0 (X × Q n ). The statement follows by Theorem 5.4 with d = 2.

5.1.
The ED polynomial of a Segre product of two projective varieties. The stabilization behavior highlighted in Corollary 5.6 has an interesting counterpart related to the ED degree and the ED polynomial of the Segre product between a projective variety and a projective space.
For a polynomial function f on a complex vector space V , we denote by V(f ) the variety defined by the vanishing of f . Let V 1 and V 2 be two complex vector spaces equipped with (real) quadratic forms q 1 , q 2 . As in §4, V 1 and V 2 are complexifications of two real Euclidean spaces. Let n i + 1 = dim(V i ) and denote Q 1 and Q 2 the isotropic quadric cones defined by the vanishing of q 1 and q 2 , i.e. Q i (x) = V(q i (x, x)).
As above, V := V 1 ⊗ V 2 itself is equipped with an Euclidean structure given by the Frobenius inner product q := q 1 ⊗q 2 , see Definition 4.1. Note that this can be regarded as the familiar space of matrices. Denote by Q the induced isotropic quadric in V 1 ⊗ V 2 .
Consider two affine cones X 1 ⊂ V 1 and X 2 ⊂ V 2 . We also denote by X 1 and X 2 the corresponding projective varieties in P(V 1 ) and P(V 2 ). Let X 1 × X 2 be their Segre product, see Definition 5.1.
A consequence of [OS,Corollary 5.5] is the following result.
A similar argument used in the proof of [Sod20, Proposition 5.2.6] leads to the following inclusion.
Proposition 5.10. The following inclusion holds true: In other words, if t ∈ V 1 ⊗ V 2 admits strictly less critical points than N , then it is forced to have a specific isotropic structure.
Summing up, we may write the extreme coefficients of EDpoly (X 1 ×X 2 ) ∨ ,t (ε 2 ) as c 0 = f 2 g α , c N = h β , for some square-free polynomials f , g, h, where V(f ) = (X 1 × X 2 ) ∨ , whenever the varieties on the right-hand side are hypersurfaces. (The polynomials are set to be 1 if the corresponding varieties have higher codimensions.) Note that when X 1 = V 1 and X 2 = V 2 we are looking at the distance function from a Segre product of projective spaces. An immediate consequence of the Eckart-Young Theorem tells us that (assuming n 1 ≤ n 2 ) EDpoly (P(V 1 )×P(V 2 )) ∨ ,t (ε 2 ) = det(t t T − ε 2 I n 1 ) .
In particular, EDpoly (P(V 1 )×P(V 2 )) ∨ ,t (ε 2 ) is a monic polynomial, i.e. the exponent β of h is zero. On the other hand, the lowest coefficient is det(t t T ). When n 1 = n 2 , then det(t t T ) = det(t) 2 = f 2 , whereas g = 1 because its corresponding variety is not a hypersurface. Otherwise n 1 < n 2 and then det(t t T ) = g, whereas in this case f = 1 because (P(V 1 ) × P(V 2 )) ∨ is not a hypersurface. That means that the exponent of g is α = 1.
These observations lead to the following more general conjecture, confirmed by experimental data from the software Macaulay2.
The validity of Conjecture 5.11 implies the stabilization of the ED degree of X × P(V 2 ) for n 2 increasing.
The previous result proves a stabilization property of the ED degree of the Segre product X × P(V 2 ). Furthermore, some experiments with the software Macaulay2 suggest the following conjecture which is a somewhat more general version of Conjecture 4.18.
Conjecture 5.13. Let X ⊂ P(V 1 ) be a projective hypersurface such that X ∩ Q 1 is reduced. Consider the Segre product X × P(V 2 ). Then where L j ⊂ P(V 1 ) is a general subspace of codimension j.