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Received: 30 January 2017 / Revised: 3 November 2017 / Accepted: 7 November 2017

Secant Degeneracy
Index of the Standard Strata in The Space of Binary Forms

Gleb Nenashev
Department of Mathematics

Stockholm University

10691 Stockholm Sweden

nenashev@math.su.se, Boris Shapiro Department of Mathematics

Stockholm University

10691 Stockholm Sweden

shapiro@math.su.se, Michael Shapiro Department of Mathematics

Michigan State University

East Lansing MI 48824-1027 USA

mshapiro@math.msu.edu

Stockholm University

10691 Stockholm Sweden

nenashev@math.su.se, Boris Shapiro Department of Mathematics

Stockholm University

10691 Stockholm Sweden

shapiro@math.su.se, Michael Shapiro Department of Mathematics

Michigan State University

East Lansing MI 48824-1027 USA

mshapiro@math.msu.edu

The space $ Pol_d\simeq \bC P^d$ of all complex-valued binary forms of degree $ d$
(considered up to a constant factor) has a standard stratification, each stratum
of which contains all forms whose set of multiplicities of their distinct roots is
given by a fixed partition $ \mu \vdash d$. For each such stratum $ S_\mu,$ we
introduce its secant degeneracy index $ \ell_\mu$ which is the minimal number
of projectively dependent pairwise distinct points on $ S_\mu$, i.e., points
whose projective span has dimension smaller than $ \ell_\mu-1$. In what follows,
we discuss the secant degeneracy index $ \ell_\mu$ and the secant degeneracy
index $ \ell_{{{\bar{\mu}}}}$ of the closure $ {{\bar{S}}}_\mu$.

Below by a form we will always mean a binary form . The standard stratification of the $ d$-dimensional projective space $ Pol_d$ of all complex-valued binary forms of degree $ d$ (considered up to a non-vanishing constant factor) according to the multiplicities of their distinct roots is a well-known and widely used construction in mathematics (see e.g. [Arnol'd2014], [Vassiliev1992], [Khesin and Shapiro1992]). Its strata denoted by $ S_\mu$ are enumerated by partitions $ \mu\vdash d$. In particular, cohomology of $ S_\mu$ with different coefficients appears in many topological problems and was intensively studied over the years, see e.g. [Vassiliev1992] and references therein.

Observe that if a quasi-projective variety $ V$ is contained in quasi-projective $ W$, then $ \ell_V\ge \ell_W$. For a positive-dimensional quasi-projective variety $ V\subset \bC P^d$, denote by $ \ell_{{{\bar{V}}}}$ the secant degeneracy index of the closure $ {{\bar{V}}} \subset \bC P^d$. Obviously, $ \ell_{{{\bar{V}}}} \le \ell_{V}$. The latter inequality can be strict as shown by Example 1 below.

The principal question considered in the present paper is as follows.

For a given partition $ \mu$, the equation

\begin{eqnarray}\label{eq:SFMU} f_{1}+f_2+\dots +f_{ \ell_{\mu}}=0, \end{eqnarray} | (1) |

Analogously, for a given partition $ \mu$, the equation

\begin{eqnarray}\label{eq:SFMUb} f_{1}+f_2+\dots +f_{ \ell_{{{\bar{\mu}}}}}=0, \end{eqnarray} | (2) |

Most of our results deal with the secant degeneracy index $ \ell_{\mu}$. However, the second part of Theorem 1 provides a non-trivial lower bound for $ \ell_{{{\bar{\mu}}}}$ generalizing a similar result of a well-known paper [Newman and Slater1979] from 1979 where the special case of partitions with equal parts was considered.

The first result of this note is as follows. Recall the notion of the refinement partial order $ ``\succ''$ on the set of all partitions of a given positive integer $ d$. Namely, $ \mu^\prime \succ \mu$ in this order if $ \mu^\prime$ is obtained from $ \mu$ by merging of some parts of $ \mu$. The unique minimal element of this partial order is $ (1)^d,$ while its unique maximal element is $ (d)$.

For a partition $ \mu=(\mu_1\ge \mu_2 \ge \dots \ge \mu_r)$, define its jump multiset $ J_\mu$ as the multiset of all positive numbers in the set $ \{\mu_1-\mu_2,\ldots,\mu_{r-1}-\mu_r, \mu_r\}$. We denote by $ h_\mu$ the minimal (positive) jump of $ \mu$, i.e. the minimal element of $ J_\mu$, and by $ h_{{\bar{\mu}}}$ the minimal jump of all partitions $ \mu'\succeq \mu$.

To formulate further results, we divide the set of all partitions into two natural disjoint subclasses as follows. Notation. For a given partition $ \mu=(\mu_1\ge \mu_2 \ge \dots \ge \mu_r)$ and a non-negative integer $ t$, define the partition $ \mu^{\langle{t}\rangle}$ as \begin{eqnarray*}\mu^{\langle{t}\rangle} :=(\mu_1+t\ge \mu_2+t\ge \dots \ge \mu_r+t). \end{eqnarray*}

We are able to characterize these two classes in the following terms.

\begin{eqnarray}\label{eq:SFGEN} f_1+f_2+\dots +f_m=0, \end{eqnarray} | (3) |

At the moment we do not have a purely combinatorial description of partitions admitting a common radical solution. However we were able to study a somewhat stronger property.

In fact, we strongly suspect that the converse to Theorem 3 holds as well.

At the moment we can settle Conjecture 1 for a large class of partitions, but not for all partitions.

The structure of the paper is as follows. In Sect. 2, we formulate several general results about $ \ell_{\mu}$, the most interesting of them being an upper bound of $ \ell_\mu$ in terms of the minimal jump. In Sect. 3, we discuss common radical solutions of (1) and in Sect. 4, we present a number of open problems.

Given a partition $ \mu=(\mu_1\ge \mu_2 \ge \dots \ge \mu_r)$, we call $ \nu=(\mu_{i_1}\ge \mu_{i_2}\ge \dots \ge \mu_{i_s})$ where $ 1\le i_1< i_2< \dots < i_s\le r$, a subpartition of $ \mu$.

Before formulating general results about $ \ell_\mu$, let us present several concrete classes of $ \mu$ and some information about the corresponding $ \ell_\mu$.

A linear combination $ ag_1+bg_2+cg_3+dg_4$ is given by \begin{eqnarray*}R(x)(a(x-p)(x-r)+b(x-q)(x-r)+c(x-p)(x-s)+d(x-q)(x-s)),\end{eqnarray*} where $ R(x)=(x-p)^{k_1}(x-q)^{k_1}(x-r)^{k_2}(x-s)^{k_2}$. Polynomials $ (x-p)(x-r)$, $ (x-q)(x-r)$, $ (x-p)(x-s)$ and $ (x-q)(x-s)$ are linearly dependent. Thus there exist $ a,b,c,d$ such that $ ag_1+bg_2+cg_3+dg_4=0$, and hence $ \ell_{\nu}\leq 4$. ⬜

Construct $ g'$ as the product of $ g$ by $ r \ell_\mu-deg(g)$ new distinct linear forms, and set $ f'_j = f_j \cdot (g')^i$, for $ j=1,\dots, \ell_\mu$. It is easy to see that each $ f'_j$ has the root partition given by $ \mu'$. Furthermore, one has \begin{eqnarray*}f'_1+\dots+f'_{\ell_\mu}=(f_1+\dots+f_{\ell_\mu})\cdot (g')^i=0,\end{eqnarray*} hence, $ \ell_{\mu^\prime} \le \ell_{\mu}$. ⬜

Both parts of Theorem 1 are settled in a similar way. Namely, given $ \mu$, let $ \{f_1,\ldots,f_{\ell}\}$ be a collection of forms solving either (1) or (2). (In the first case $ \ell=\ell_\mu$ and in the second case $ \ell=\ell_{{{\bar{\mu}}}}$.) Assume that $ \{f_1,\ldots,f_{\ell}\}$ gives a counterexample to the statement. Denote by $ g$ the GCD of $ \{f_1,\ldots,f_{\ell}\}$ and consider the relation \begin{eqnarray*}\frac{f_1}{g}+\cdots+\frac{f_{\ell}}{g}=0.\end{eqnarray*}

In case (i) of Theorem 1, for any $ i$, every root of the polynomial $ \frac{f_i}{g}$ has multiplicity at least $ h_\mu$, because this multiplicity equals $ \mu_k-\mu_l$ for some $ k\leq l$. Observe that, for all $ k\le l$, $ \mu_k-\mu_l$ is either $ 0$ or is greater than or equal to $ h_\mu$.

In case (ii) of Theorem 1, for any $ i$, every root of the polynomial $ \frac{f_i}{g}$ has multiplicity at least $ h_{{{\bar{\mu}}}}$, because this multiplicity equals $ \sum_{k\in A} \mu_k-\sum_{l\in B}\mu_l$, where $ A$ and $ B$ are two subsets of $ \{1,\dots , r\}$. Observe that, for all pairs $ (A',B')$ such that $ A'\cap B'=\emptyset$, $ |\sum_{k\in A'} \mu_k-\sum_{l\in B'}\mu_l|$ is either $ 0$ or is greater than or equal to $ h_{{{\bar{\mu}}}}$.

The rest of the proof is the same in both cases. In what follows, $ h$ stands for $ h_\mu$ in case (i) and for $ h_{{{\bar{\mu}}}}$ in case (ii). Consider the sequence of Wronskians \begin{eqnarray*}w_i=W\left(\frac{f_1}{g}, \ldots,\frac{f_{i-1}}{g},\frac{f_{i+1}}{g},\ldots,\frac{f_{\ell}}{g}\right),\; i=1,\dots, \ell.\end{eqnarray*} All these Wronskians are proportional to each other due to the latter relation.

Let $ \alpha$ be a root of some $ f_i$. There exists an index $ s$ such that $ \frac{f_s}{g}$ is not divisible by $ (x-\alpha)$, since otherwise $ g$ is not the GCD.

For a given $ t,$ consider the multiplicity of the root of $ w_t$ at $ \al$. It satisfies the inequality: \begin{eqnarray*}ord_\alpha (w_t)\ge \sum\left(ord_\alpha\left(\frac{f_j}{g}\right)\right)-(\ell-2)\#\left\{i:(x-\alpha)|\frac{f_i}{g}\right\},\end{eqnarray*} because any column of the Wronski matrix corresponding to $ (x-\alpha)|\frac{f_j}{g}$ is divisible by $ (x-\alpha)^{ord_\alpha{\left(\frac{f_j}{g}\right)}-\ell+2}$.

Hence, \begin{eqnarray*} \deg w_1 &\ge & \sum_{i=1}^{\ell} \left(\deg \left(\frac{f_i}{g}\right)-(\ell-2)\#_{roots}\left(\frac{f_i}{g}\right)\right)\\ &=&\ell (|\mu|- \deg g )-(\ell-2)\sum_{i=1}^{\ell} \#_{roots}\left(\frac{f_i}{g}\right).\end{eqnarray*} On the other hand, \begin{eqnarray*}\deg w_1 \leq (\ell - 1)\left(\deg\left(\frac{f_i}{g}\right)-\ell +2\right)=(\ell-1) (|\mu|- \deg g)-(\ell-1)(\ell-2).\end{eqnarray*} We obtain \begin{eqnarray*}(\ell-1) (|\mu|- \deg g)-(\ell-1)(\ell-2)\ge \ell (|\mu|- \deg g)-(\ell-2)\sum_{i=1}^{\ell} \#_{roots}\left(\frac{f_i}{g}\right),\end{eqnarray*} i.e., \begin{eqnarray*}(\ell-2)\sum_{i=1}^{\ell} \#_{roots}\left(\frac{f_i}{g}\right)-(\ell-1)(\ell-2)\ge |\mu|- \deg g.\end{eqnarray*}

The number $ \#_{roots}\left(\frac{f_i}{g}\right)$ of distinct roots is at most $ \frac{|\mu|- \deg g}{h}$, because each root has multiplicity at least $ {h}$. Thus \begin{eqnarray*}(\ell-2)(\ell-1)\frac{|\mu|- \deg g}{h_\mu}-(\ell-1)(\ell-2)\ge |\mu|- \deg g.\end{eqnarray*} Hence, $ (\ell-2)(\ell-1)> {h}.$ ⬜

For the term $ f_i=c_i(x-a_{i,1})^{\mu_1}\cdots (x-a_{i,r})^{\mu_r}$, define \begin{eqnarray*}g_i:=(x-a_{i,1})^{\mu_1-m+2}\cdots (x-a_{i,r})^{\mu_r-m+2}.\end{eqnarray*} Observe that $ g_i$ is a polynomial, because any root of $ f_i$ has multiplicity at least $ \mu_r> m$.

Consider the sequence of Wronskians \begin{eqnarray*}w_i=W(f_1, \ldots,f_{i-1},f_{i+1},\cdots,f_{m}),\; i=1,\dots , m.\end{eqnarray*} They are proportional to each other, because $ f_1+\ldots+f_m=0$. Notice that, for $ i\neq t,$ the column in the Wronski matrix for $ w_t$ corresponding to $ f_i$ is divisible by $ g_i$. Hence $ w_t$ is divisible by $ {\prod_{i=1}^{m} g_i}/{g_t}$.

Since $ \{f_1,\ldots,f_{m}\}$ is not a common radical solution, there exists $ \alpha\in \bC$, such that $ \alpha$ is a root of $ f_{p}$ but is not a root of $ f_q$ for some $ p\neq q$.

Since the Wronskians $ w_p$ and $ w_q$ are proportional, they are divisible by \begin{eqnarray*}LCM\left(\frac{\prod_{i=1}^{m} g_i}{g_p},\frac{\prod_{i=1}^{m} g_i}{g_q}\right)=\frac{\prod_{i=1}^{m} g_i}{GCD(g_p,g_q)}=\frac{\prod_{i=1}^{m} g_i}{g_p}\frac{g_p}{GCD(g_p,g_q)}.\end{eqnarray*} Then these Wronskians are divisible by $ \frac{\prod_{i=1}^{m} g_i}{g_p}(x-\alpha)^{\mu_r-m+2}$. Therefore their degrees are greater than or equal to \begin{eqnarray*}(m-1)(|\mu|-r(m-2))+\mu_r-m+2.\end{eqnarray*} On the other hand, the degrees of the Wronskians are at most $ (m-1)(|\mu|-m+2).$ Thus, \begin{eqnarray*}(m-1)(|\mu|-m+2)\geq (m-1)(|\mu|-r(m-2))+\mu_r-m+2,\end{eqnarray*} which implies that $ -(m-1)(m-2)\geq -r(m-1)(m-2)+\mu_r-m+2.$ After straightforward simplifications the latter inequality gives \begin{eqnarray*}m-1 \geq \sqrt{\frac{\mu_r}{r-1}}.\end{eqnarray*} Contradiction. ⬜

Now we present a sufficient condition for $ \mu$ to have a growing secant degeneracy index.

We continue with the proof of Theorem 3.

For any $ \pi \in \mathcal{D}_{(a_1,\dots,a_r)},$ define $ {\hat{f}}_\pi:=\frac{f_\pi}{g}\in S_{|\mu|-\sum_{i=1}^r{a_i}}$. If $ |\mathcal{D}_{(a_1,\dots,a_r)}|\geq |\mu|-\sum_{i=1}^r{a_i}+2$, then the forms $ {\hat{f}}_\pi $ are linearly dependent. Therefore, the forms $ f_\pi$, where $ \pi$ runs over $ \mathcal{D}_{(a_1,\dots,a_r)}$, are also linearly dependent. ⬜

The next proposition shows that if there are many jumps of small sizes, then the secant degeneracy index is bounded.

Consider the set of permutations of $ \mu=\{\mu_1,\ldots,\mu_r\}$ such that:

- $ i\notin \{j_1,\dots,j_t\}$, $ \pi_i\geq \mu_i$;
- $ i\in \{j_1,\dots,j_t\}$, $ \pi_i\geq \mu_{i+1}$.

The number of such permutations is $ 2^t$. By Theorem 3, there is a solution of the size $ \sum_{i=1}^t(\mu_{j_i}-\mu_{j_i+1})+2\leq d\cdot t + 2$, because \begin{align*} {d\cdot t+2 \leq d\cdot(\log d+\log \log d+2)+2\leq d\cdot (\log d+\log \log d+3 )\\ \qquad\qquad\quad \leq d\cdot 2\cdot \log d\leq 2^{\log d+\log \log d+1}\leq 2^{t}} \end{align*} which finishes the proof. ⬜

It is rather obvious that all partitions with two parts have growing secant degeneracy index. Indeed, if there exists a common radical solution of (3) , then its length $ m$ is smaller than or equal to $ r!$ which in case of two parts equals $ 2$.

- (i) For partitions $ \mu=(p,2)$, one has that $ \ell_\mu=3$ when $ p=2,$ and $ \ell_\mu=4$ when $ p> 2$.
- (ii) For partitions $ \mu=(p,3)$, one has that $ \ell_\mu=4$ when $ p=3,4,5$ and $ \ell_\mu=5$ when $ p> 7$. Cases $ \mu=(6,3)$ and $ \mu=(7,3)$ are still open.

In case $ \mu=(5,3)$ we found an example: \begin{eqnarray*}f_1+f_2-f_3-f_4=0,\end{eqnarray*} where $ f_1(x)=(x+c_1^5y)^3(x+c_1^{-3}y)^5$; $ f_2(x)=(x+c_2^5y)^3(x+c_2^{-3}y)^5$; $ f_3(x)=(x+c_1^{-5}y)^3(x+c_1^{3}y)^5$; $ f_4(x)=(x+c_2^{-5}y)^3(x+c_2^{3}y)^5$. Here \begin{eqnarray*}c_1=-c_2= \left( \frac{1+i\sqrt{35}}{6} \right)^{\frac{1}{4}}.\end{eqnarray*} ⬜

Using computer algebra packages we were able to prove the following statement.

- (i) $ \mu$ has a stabilising secant degeneracy index;
- (ii) $ \mu$ has a strongly stabilising secant degeneracy index;
- (iii) the triple $ a\le b \le c$ belongs to one of the following three types: $ a=b$, $ c=a+1$; $ b=a+1$, $ c=a+2$; $ b=a+1$, $ c=a+3$.

- 1. Conjecture 1 from the introduction is equivalent to the claim that for an appropriate choice of an $ r$-tuple of distinct complex numbers, a certain matrix whose entries are polynomials in these numbers has full rank. In other words, that some maximal minor of this matrix is a non-trivial polynomial in the latter $ r$-tuple which is highly plausible. Unfortunately, the structure of the matrix is rather involved and so far we are only able to handle a large number of special cases.
- 2. Conditions formulated in Proposition 2 are difficult to check which motivates the following questions.

- 3. The next statement is obvious.

Based on a substantial number of calculations, we conjecture the following.

[Arnol'd2014] Arnol'd, V.: The cohomology ring of the colored braid group.
Collect. Works 2
, 183–186 (2014)

[Beltrametti and Sommese1995] Beltrametti, M.C., Sommese, A.J.: The adjunction theory of complex projective varieties. de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1995)

[Chiantini and Ciliberto2010] Chiantini, L., Ciliberto, C.: On the dimension of secant varieties. J. Eur. Math. Soc. (JEMS) 12 (5), 1267–1291 (2010)

[Chiantini et al.2015] Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: On generic identifiability of symmetric tensors of subgeneric rank, arXiv:1504.00547v2, April (2015)

[Eisenbud and Harris1987] Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3–13, In: Proceedings of Symposia in Pure Mathematics, 46, Part 1, American Mathematical Society, Providence, RI, (1987)

[Khesin and Shapiro1992] Khesin, B., Shapiro, B.: Swallowtails and Whitney umbrellas are homeomorphic. J. Alg. Geom. 1 , 549–560 (1992)

[Newman and Slater1979] Newman, D.J., Slater, M.: Waring's problem for the ring of polynomials. J. Number Theory 11 (4), 477–487 (1979)

[Reznick1999] Reznick, B.: Linearly dependent powers of quadratic forms, Preliminary report: 1999–2011, slides. . Accessed 2 Dec 2017

[Vassiliev1992] Vassiliev, V.: Complements of discriminants of smooth maps: topology and applications. Translated from the Russian by B. Goldfarb. Translations of mathematical monographs, 98. American Mathematical Society, Providence, RI, vi+208 pp(1992)

[Beltrametti and Sommese1995] Beltrametti, M.C., Sommese, A.J.: The adjunction theory of complex projective varieties. de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1995)

[Chiantini and Ciliberto2010] Chiantini, L., Ciliberto, C.: On the dimension of secant varieties. J. Eur. Math. Soc. (JEMS) 12 (5), 1267–1291 (2010)

[Chiantini et al.2015] Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: On generic identifiability of symmetric tensors of subgeneric rank, arXiv:1504.00547v2, April (2015)

[Eisenbud and Harris1987] Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3–13, In: Proceedings of Symposia in Pure Mathematics, 46, Part 1, American Mathematical Society, Providence, RI, (1987)

[Khesin and Shapiro1992] Khesin, B., Shapiro, B.: Swallowtails and Whitney umbrellas are homeomorphic. J. Alg. Geom. 1 , 549–560 (1992)

[Newman and Slater1979] Newman, D.J., Slater, M.: Waring's problem for the ring of polynomials. J. Number Theory 11 (4), 477–487 (1979)

[Reznick1999] Reznick, B.: Linearly dependent powers of quadratic forms, Preliminary report: 1999–2011, slides. . Accessed 2 Dec 2017

[Vassiliev1992] Vassiliev, V.: Complements of discriminants of smooth maps: topology and applications. Translated from the Russian by B. Goldfarb. Translations of mathematical monographs, 98. American Mathematical Society, Providence, RI, vi+208 pp(1992)

- × Arnol'd, V.: The cohomology ring of the colored braid group. Collect. Works 2 , 183–186 (2014)
- × Beltrametti, M.C., Sommese, A.J.: The adjunction theory of complex projective varieties. de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1995)
- × Chiantini, L., Ciliberto, C.: On the dimension of secant varieties. J. Eur. Math. Soc. (JEMS) 12 (5), 1267–1291 (2010)
- × Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: On generic identifiability of symmetric tensors of subgeneric rank, arXiv:1504.00547v2, April (2015)
- × Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3–13, In: Proceedings of Symposia in Pure Mathematics, 46, Part 1, American Mathematical Society, Providence, RI, (1987)
- × Khesin, B., Shapiro, B.: Swallowtails and Whitney umbrellas are homeomorphic. J. Alg. Geom. 1 , 549–560 (1992)
- × Newman, D.J., Slater, M.: Waring's problem for the ring of polynomials. J. Number Theory 11 (4), 477–487 (1979)
- × Reznick, B.: Linearly dependent powers of quadratic forms, Preliminary report: 1999–2011, slides. . Accessed 2 Dec 2017
- × Vassiliev, V.: Complements of discriminants of smooth maps: topology and applications. Translated from the Russian by B. Goldfarb. Translations of mathematical monographs, 98. American Mathematical Society, Providence, RI, vi+208 pp(1992)

Definition 1 Remark 1
Remark 2
Problem 1
Example
1 Theorem
1 Definition
2 Definition
3 Proposition
2 Definition
4 Theorem
3 Conjecture
1

2. General Results on the Secant Degeneracy Index
Proposition 4 Example
2 Proposition
5 Definition
5 Proposition
6 Corollary
1

3. Partitions with Growing and Stabilising Secant Degeneracy
Index
Theorem 7 Corollary
2 Remark 3
Corollary
3 Proposition
8
3.1. Examples

4. Final Remarks and Problems
Proposition 9 Proposition
10

Problem 2 Problem 3
Proposition
11 Conjecture
2