Algebraic boundaries among typical ranks for real binary forms of arbitrary degree

We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci.


Introduction
Let f ∈ R d = K[x, y] d be a binary form of degree d, where K = R or C. By definition, see e.g. [Lan12], the K-rank of f is the minimum integer r such that f admits a decomposition f = r i=1 α i (ℓ i ) d where α i ∈ K and ℓ i ∈ K[x, y] 1 for i = 1, . . . , r. If K = C, the generic rank, i.e., the complex rank of a general complex binary form of degree d, is ⌈ d+1 2 ⌉. In the case K = R, the notion of generic rank is replaced by the notion of typical ranks. A rank is called typical for real binary forms of degree d if it occurs in an open subset of R d , with respect to the Euclidean topology. By [Ble15] it is known that a rank r is typical if and only if d+1 2 ≤ r ≤ d. Denoting by R d,r the interior of the semi-algebraic set {f ∈ R d : rk R (f ) = r} in the real vector space R d , then a rank r is typical exactly when R d,r is not empty.
Let us now assume d+1 2 ≤ r ≤ d. Following [LS16] and [BS18], we define the topological boundary ∂(R d,r ) as the set-theoretic difference of the closure of R d,r and the interior of the closure of R d,r . Thus, if f ∈ ∂(R d,r ) then every neighborhood of f contains a generic form of real rank equal to r and also a generic form of real rank different from r. We have that ∂(R d,r ) is a semi-algebraic subset of R d of pure codimension one. We define the algebraic boundary ∂ alg (R d,r ), also called real rank boundary, as the Zariski closure of the topological boundary ∂(R d,r ) in the complex projective space P(C[x, y] d ).
The algebraic boundary for maximum rank r = d coincides with the discriminant hypersurface. Indeed by [CO12] and [CR11], we know that the open set R d,d corresponds to the locus of real-rooted forms, that is forms with all distinct and real roots. In the opposite case, the algebraic boundary for minimum rank r = ⌈ d+1 2 ⌉ has been described in [LS16]. It is irreducible when d is odd, and it has two irreducible components when d is even. From these general results, it follows a complete description of all the algebraic boundaries with low degree d ≤ 6.
In [BS18], we completely described the algebraic boundaries for the next two cases, d = 7 and d = 8. More precisely, we show in [BS18] that all the boundaries between two typical ranks are unions of dual varieties to suitable coincident root loci. Coincident root loci are well-studied varieties which parametrize binary forms with multiple roots, see Section 1.1 for the precise definitions.
In this paper we study the algebraic boundaries for forms of arbitrary degree, and our main result is the following: Theorem 0.1 (Theorems 3.1 and 3.2). For any degree d and any typical rank d+1 2 ≤ r ≤ d, the algebraic boundary ∂ alg (R d,r ) is a union of dual varieties to coincident root loci.
We remark finally that the study of algebraic boundaries for forms with more than two variables is a challenging and quite open problem, see [MMSV17,Ven19].
The paper is organized as follows: in the preliminary Section 1 we recall some basic notions and results about coincident root loci, higher associated subvarieties and apolar maps. Section 2 is devoted to the detailed analysis of the pullbacks, via apolar maps, of higher associated varieties to coincident root loci. The main result of this section is Theorem 2.1, whose two corollaries (Corollaries 2.3 and 2.4) are key tools in the proof of Theorem 0.1. In Section 3 we prove Theorem 0.1: more precisely, we consider the case of odd degree in Theorem 3.1, and the case of even degree in Theorem 3.2.
1. Preliminary 1.1. Coincident root loci. Let r be a positive integer. A partition of r is an equivalence class, under reordering, of lists of positive integers λ = [λ 1 , . . . , λ n ] such that n i=1 λ i = r. We denote by |λ| the length n of the partition. Alternatively, the partition λ can be represented by the list of integers m 1 , . . . , m k defined as m j = |{i : λ i = j}|, and clearly k j=1 jm j = r. Given a partition λ as above, the coincident root locus ∆ λ ⊂ P r = P(C[x, y] r ) associated with λ is the set of binary forms f of degree r which admit a factorization f = n i=1 ℓ λ i i for some linear forms ℓ 1 , . . . , ℓ n ∈ C[x, y] 1 . These varieties have been extensively studied, see e.g. [Wey89,Kat03,Chi03,Chi04,Kur12].
We have a unirational parameterization of degree m 1 !m 2 ! · · · m k !: In particular, the dimension of ∆ λ is n. The degree of ∆ λ was determined by Hilbert [Hil87] . He showed that If λ and µ are two partitions of r, we have ∆ µ ⊆ ∆ λ if and only if λ is a refinement of µ (equiv., µ is a coarsening of λ). In [Chi03] and subsequently in [Kur12] it has been shown that the singular locus Sing(∆ λ ) is given by the union of ∆ µ for some suitable coarsenings µ of λ (see [Chi03,Definition 5.2] and [Kur12, Proposition 2.1] for the precise description). In particular, one has that ∆ λ is smooth if and only if λ 1 = · · · = λ n . Otherwise the singular locus is of codimension 1.
The dual variety (∆ λ ) ∨ of ∆ λ ⊂ P(C[x, y] r ) lies in the projective space P(C[∂ x , ∂ y ] r ), and it is shown in [LS16] (see also [Kat03]) that (∆ λ ) ∨ ⊂ P(C[∂ x , ∂ y ] r ) is a subvariety of codimension m 1 , which is given by the join of the (n − m 1 ) coincident root loci ∆ (d−λ i +2,1 λ i −2 ) for 1 ≤ i ≤ n with λ i ≥ 2. In particular, ∆ ∨ λ is a hypersurface if and only if λ i ≥ 2 for all i. In this case its degree has been computed in [Oed12] and it is 1.2. Higher associated subvarieties. Let G(l, r) = G(l, P r ) denote the Grassmannian of l-dimensional projective subspaces of P r . Let X ⊂ P r C be an irreducible projective variety of dimension n. Recall that the j-th higher associated variety CH j (X) to X is the closure of the set of the (r − n − 1 + j)-dimensional subspaces H ⊂ P r such that H ∩ X = ∅ and dim(H ∩ T x X) ≥ j for some smooth point x ∈ H ∩ X, see [GKZ08].
For our purpose it is useful to consider a natural generalization of higher associated subvarieties, which we now introduce. For any integers j, l with 0 ≤ j ≤ l ≤ r and j ≤ n, we define Proposition 1.1. The scheme I l j is smooth and irreducible of dimension . Proof. We have the following diagram of natural projections: For each x ∈ X \ Sing(X), we have q −1 1 (x) ≃ {B ∈ G(j, T x X) : x ∈ B} ≃ G(j − 1, n − 1). Thus I j j is smooth and irreducible of dimension , and the claim of the proposition follows.
Definition 1.2. Let I l j be the closure of I l j in X × G(j, P r ) × G(l, P r ), and denote by π 2 : I l j → G(l, P r ) the last projection. The scheme π 2 (I l j ) = π 2 (I l j ) ⊂ G(l, P r ) will be denoted by Ξ l j = Ξ l j (X). Remark 1.3. It follows from the definition and Proposition 1.1 that we have . When j = 0 and l < r − n we have that π 2 is birational. In particular, the above inequality is an equality, that is is the Hurwitz hypersurface (see [Stu17]); and Ξ r−1 n (X) ⊂ G(r − 1, P r ) is the dual variety of X.
Let us recall that CH j (X) has codimension one in G(r − n − 1 + j, P r ) if and only if 0 ≤ j ≤ dim(X) − def(X), where def(X) denotes the dual defect of X (see [GKZ08] and [Koh16]).
Example 1.5. Let P = (∅ ⊂ P 0 ⊂ · · · ⊂ P r−1 ⊂ P r = P r ) be a nested sequence of projective subspaces, with dim P i = i, and let a = (a 0 , . . . , a l ) be a sequence of integers with r − l ≥ a 0 ≥ · · · ≥ a l ≥ 0. Then the so-called Schubert variety Σ a (P) ⊂ G(l, r) coincides with the following intersection: 1.3. Apolar maps and apolarity. Let K ⊆ C be a field. Let R = K[x, y] be a polynomial ring and let D = K[∂ x , ∂ y ] be the dual ring of differential operators. The ring D acts on R with the usual rules of differentiations, and we have the pairing the apolar ideal f ⊥ ⊂ D is given by all the operators which annihilates f , that is For instance, if l = ax + by ∈ R 1 , then l ⊥ is generated by the operator −b∂ x + a∂ y ∈ D 1 . The which in the standard basis is represented by the catalecticant (or Hankel) matrix (up to multiplying the rows by scalars), see e.g. [ER93]: For a general form f ∈ R d , with d ≥ r ≥ d − r, the matrix A d,r has maximal rank, and hence dim ker(A d,r ) = 2r − d. Thus we have a rational map, called the apolar map, In coordinates the map Ψ d,r is defined by the maximal minors of the matrix A d,r . For d = r, the map Ψ d,d gives an identification between P d = P(R d ) and is a birational map onto its image. This implies that we can recover a general binary form of degree d from the component (f ⊥ ) r ⊂ f ⊥ . A more precise result is the following (see e.g. [IK99]): Proposition 1.6. Assume that f ∈ R d is not a power of a linear form. Then its apolar ideal f ⊥ is generated by two forms g, g ′ such that d = deg g + deg g ′ − 2 and gcd(g, g ′ ) = 1. Conversely, any two such forms generate an ideal f ⊥ for some projectively unique f ∈ R d . We . The forms that are not generated in generic degrees form a subvariety of R d , which has codimension 1 if d is even and codimension 2 if d is odd. Indeed this subvariety is defined by the maximal minors of the intermediate catalecticant matrix A classical result is the following: It follows that a form f has rank less than or equal to r if and only if (f ⊥ ) r = f ⊥ ∩ D r contains a form with all roots distinct and in K. Recall that when K = R, such a form is called real-rooted.

Pullbacks of higher associated varieties to coincident root loci
In this section we study the geometry of the pullbacks, via apolar maps, of higher associated varieties to coincident root loci, Ψ −1 d,r (CH j (∆ λ )). This analysis is a key tool in the description of the algebraic boundaries that we will carry out in Section 3, and we think it is interesting in itself.
Let r be a positive integer, and let λ = [λ 1 , . . . , λ n ] be a partition of r of length |λ| = n. Consider for any integer 0 ≤ j ≤ min{n, r − n − 1} the following set of partitions and let us fix λ ′ ∈ D j (λ). Let ∆ λ ⊂ P r (resp. ∆ λ ′ ⊂ P r−j ) be the coincident root locus corresponding to the partition λ (resp. λ ′ ). Let d = r + n − j and consider the apolar maps: We consider the j-th higher associated variety of ∆ λ ⊂ P r , which is a hypersurface if and only if j ≤ n − m 1 (λ), where m 1 (λ) is the number of 1 in the partition. Set l = |λ ′ | ≤ n and consider also the irreducible variety associated to Theorem 2.1. Let notation be as above. Then the following formula holds settheoretically: Thus there exist J + 1 linearly independent forms q 0 , . . . , q J of degree n such that Consider now the form p = ℓ λ 1 −1 1 · · · ℓ λn−1 n (f ), which has degree d − r + n = 2n − j. Of course q 0 , . . . , q J annihilates p, yielding that the dimension of (p ⊥ ) n = P(ker A 2n−j,n p ) is at least J, where A 2n−j,n p is the catalecticant matrix of p of size (n − j + 1) × (n + 1). This means that the rank of A 2n−j,n p is at most n − J.
We now show the inclusion ⊇.
Let us consider the j-dimensional linear subspace L = {qh : h binary form of degree j} ⊆ (f ⊥ ) r ⊂ P r .
We then have the following: Corollary 2.4. Let notation be as above. If j ≤ n − m 1 (λ), then the following formula holds set-theoretically: Proof. Since j ≤ n − m 1 (λ), then the left-hand side of (2.2) is a hypersurface. Hence we can exclude from the right-hand side all the components of codimension higher than 1. By Remark (2.2), this corresponds to ask that |λ ′ | = n, and in this case we also have Ξ r−j−n−1 2.1. On the multiplicities of the components. Here we try to give a more precise geometric description of the pullbacks, via apolar maps, of higher associated varieties to coincident root loci. In particular we study the multiplicity of the components which appear in Equation (2.2) of Theorem 2.1.
Based on a number of experimental verifications and some considerations on the singular loci of the higher associated varieties, we formulate here the following: Conjecture 2.5. If j ≤ n−m 1 (λ), then the following formula holds scheme-theoretically: We now illustrate the formula above with some few examples.

Algebraic boundaries among typical ranks for binary forms
Now, thanks to the results of the previous section, we are able to describe all the algebraic boundaries for binary forms of arbitrary degree d.
We will consider the case of odd degree in Theorem 3.1, and the case of even degree in Theorem 3.2.
Theorem 3.1. If d = 2k − 1 is odd, then the algebraic boundaries for degree d binary real forms satisfy the following: (1) If i = 0 then ∂ alg (R d,k+i ) = (∆ (3,2 k−2 ) ) ∨ . ( where ρ runs among the partitions of d with all parts greater than or equal to 2 and of length between k − 1 − i and k − 1; in particular, Proof. Part (3) has been shown in [CR11], and part (1) is one of the results contained in [LS16]. Now we apply the idea, given in [LS16] and developed in [BS18], to study the boundary ∂ alg (R d,k+i ) between R d,k+i and R d,k+i+1 ∪ · · · ∪ R d,d , where 1 ≤ i ≤ k − 2.
Let {f ε } ε be a continuous family of forms going from R d,k+i to R d,k+i+1 ∪ · · · ∪ R d,d and crossing some irreducible component of the boundary ∂ alg (R d,k+i ) at a general point f 0 = f . Thus, we assume that f −ε ∈ R d,k+i and f ε ∈ R d,k+i+1 ∪ · · · ∪ R d,d for any small ε with ε > 0.
Let G(2i, k + i) be the Grassmannian of 2i-planes in P(D k+i ), and consider the apolar map which, since i ≥ 1, is a birational map onto its image Z d,k+i ⊂ P ( k+i+1 2i+1 )−1 . The exceptional locus of Ψ d,k+i is contained in the locus of non generated in generic degree forms, which has codimension 2 in P d . Therefore we can assume that Ψ d,k+i is a local isomorphism at f ε for each ε, and hence it sends bijectively the family {f ε } ε into a continuous family {Π ε } ε of apolar 2i-planes.
From the Apolarity Lemma 1.7, we obtain that the 2i-plane Π ε , with ε < 0, contains a real-rooted form h ε , while Π ε , with ε > 0, does not contain any real-rooted form. The set of real-rooted forms is a full-dimensional connected semi-algebraic subset of P k+i , and the Zariski closure of its topological boundary is the discriminant hypersurface ∆ = ∆ (2,1 k+i−2 ) . Remind also that the singular locus of a coincident root locus ∆ λ is given by a union of ∆ µ , for suitable coarsenings µ of λ. Thus the limit h 0 = lim ε→0 − h ε must belong to ∆, and moreover there must exist a ∈ {0, . . . , 2i} such that h 0 is a (smooth) point of a coincident root locus ∆ ′ ⊂ P k+i corresponding to a partition λ of k + i of length k + i − a − 1 and the tangent space T h 0 (∆ ′ ) intersects Π 0 ≃ P 2i in a subspace H of dimension 2i − a passing through h 0 . This implies that Π 0 ∈ CH 2i−a (∆ ′ ), so that f ∈ Ψ −1 d,k+i (CH 2i−a (∆ ′ )). Thus we have that each irreducible component of the boundary that separates R d,k+i from R d,k+i+1 ∪ · · · ∪ R d,d is contained in the following union where λ runs among all the partitions of k + i of length k + i − a − 1, and where moreover Ψ −1 d,k+i (CH 2i−a (∆ λ )) has codimension 1. By Corollary 2.3, this last condition is equivalent to the fact that CH 2i−a (∆ λ ) is a hypersurface in G(2i, k + i), that is such that 2i − a ≤ k + i − a − 1 − m 1 (λ). Thus, by applying Corollary 2.4, we deduce that 2i a=0 λ where µ runs among the partitions of k − i + a of length k + i − a − 1, ν runs among the partitions of d of length k + i − a − 1 with parts ≥ 2, and ρ runs among the partitions of d of length between k − 1 − i and k − 1 + i with parts ≥ 2. Of course we must have a ≥ i and the length of a partition ρ is at most k − 1. Let where λ runs among the partitions of d of length k − 1 − i and with all parts ≥ 2. We have shown that and now the proof follows by an easy induction on i.
Proof. Thanks to [LS16] and [CR11] we have only to show part (2). The proof for this part is quite similar to that of Theorem 3.1 and we now sketch it. Consider a continuous family of degree d forms {f ε } ε such that f −ε ∈ R d,k+i and f ε ∈ R d,k+i+1 ∪ · · · ∪ R d,d for any small ε with ε > 0, and where we can also require that f ε / ∈ (∆ 2 k ) ∨ . The birational map has exceptional locus contained in the hypersurface (∆ 2 k ) ∨ , so it sends bijectively the family {f ε } ε into a continuous family of apolar (2i − 1)-planes {Π ε }. From the Apolarity Lemma 1.7, we deduce that f = f 0 must belong to the following union where λ runs among all the partitions of k + i of length k + i − a − 1 and such that Ψ −1 d,k+i (CH 2i−a−1 (∆ λ )) has codimension 1. Now we can conclude by applying Corollaries 2.3 and 2.4, as in the proof of Theorem 3.1.
Furthermore, notice that parts (2) of Theorems 3.1 and 3.2 are not sharp. Indeed for low degrees, d ≤ 8, we know by [BS18] that the formula holds with equality when we take only partitions with length ρ ∈ {k − i − 1, k − i} if d is odd, and with length ρ ∈ {k − i, k − i + 1} if d is even. We expect that the same happens in general. This also would imply that an algebraic boundary exists only between a region R d,r and R d,r+1 (or R d,r−1 ).
In Tables 1 and 2, we report the number of partitions of given length of some integers, in accordance to the formulas of Theorems 3.1 and 3.2. The sum of the elements of the k-th row up to position i gives an upper bound for the number of components of the algebraic boundary ∂ alg (R d,k+i ), where d = 2k − 1 if d is odd, and d = 2k if d is even. However, by taking into account our expectations described in Remark 3.3, a finer upper bound for the number of components of ∂ alg (R d,k+i ) would be given just by the sum of the two numbers in the k-th row corresponding to the position i and i − 1.