Related papers: Strength and slice rank of forms are generically e…

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a…

This paper investigates defining equations for secant varieties of the variety of reducible polynomials, which geometrically encode the notions of strength and slice rank of homogeneous polynomials. We present three main results. First, we…

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop…

We give the stratification by the symmetric tensor rank of all degree $d \ge 9$ homogeneous polynomials with border rank $5$ and which depend essentially on at least 5 variables, extending previous works (A. Bernardi, A. Gimigliano, M.…

The strength of a homogeneous polynomial (or form) is the smallest length of an additive decomposition expressing it whose summands are reducible forms. Using polynomial functors, we show that the set of forms with bounded strength is not…

We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous…

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is…

We prove that a generic homogeneous polynomial of degree $d$ is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order $k$ whenever $k\leq\frac{d}{2}-1$.

Here we study the typical rank for real bivariate homogeneous polynomials of degree $d\ge 6$ (the case $d\le 5$ being settled by P. Comon and G. Ottaviani). We prove that $d-1$ is a typical rank and that if $d$ is odd, then $(d+3)/2$ is a…

Let $f$ be a homogeneous polynomial over a field. For many fields, including number fields and function fields, we prove that the strength of $f$ is bounded above by a constant multiple of the Birch rank of $f.$ The constant depends only on…

We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent…

We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd…

In this paper, the results in [Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204], for polynomial Hessians with determinant zero in small dimensions $r+1$, are generalized to similar results in arbitrary dimension, for polynomial…

We show that a real homogeneous polynomial f(x,y) with distinct roots and degree d greater or equal than 3 has d real roots if and only if for any (a,b) not equal to (0,0) the polynomial af_x+bf_y has d-1 real roots. This answers to a…

For fixed degree and increasing number of variables the dimension of the vector space of $n$-variate real symmetric homogeneous polynomials (forms) of degree $d$ stabilizes. We study the limits of the cones of symmetric nonnegative…

Any homogeneous harmonic polynomial can be decomposed as a sum of powers of isotropic linear forms, that is, linear forms whose coefficients are the coordinates of isotropic points. The minimum size of such decompositions for a harmonic…

We exhibit, for each even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives the lower bound…

Let $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < \binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\times ... \times (m+1)$, $d$ times)…

We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree…

We introduce the strength for sections of a line bundle on an algebraic variety. This generalizes the strength of homogeneous polynomials that has been recently introduced to resolve Stillman's conjecture, an important problem in…