Related papers: Progress on the symmetric Strassen conjecture

We prove the $P=W$ conjecture for $\mathrm{GL}_n$ for all ranks $n$ and curves of arbitrary genus $g\geq 2$. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the…

Let $\mu_n$ be the standard Gaussian measure on $\mathbb{R}^n$ and $X$ be a random vector on $\mathbb{R}^n$ with the law $\mu_n$. U-conjecture states that if $f$ and $g$ are two polynomials on $\mathbb{R}^n$ such that $f(X)$ and $g(X)$ are…

We revisit a geometric lower bound for Waring rank of polynomials (symmetric rank of symmetric tensors) of Landsberg and Teitler and generalize it to a lower bound for rank with respect to arbitrary varieties, improving the bound given by…

Using the theory of spherical varieties and especially Frobenius splitting results for symmetric varieties, we give a type independent very short proof of Wahl's conjecture for cominuscule homogeneous spaces for all primes different from 2.

In \cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$ and the rank of $(G,\cdot)$ is smaller than $p-1$, then the order of any element is the same in the additive and…

Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's…

For graphs F and G an F-matching in G is a subgraph of G consisting of pairwise vertex disjoint copies of F. The number of F-matchings in G is denoted by s(F,G). We show that for every fixed positive integer m and every fixed tree F, the…

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…

This work can be thought as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two…

We extend several predecessor works on even sextic monogenic polynomials. In particular, we prove a conjecture of Lenny Jones, thereby classifying even sextic monogenic polynomials with cyclic Galois group. This result is key to completing…

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions,…

In this paper we announce a conjecture concerning enumeration of n-times persymmetric matrices over F_2 by rank. To justify our statement we remark that the formulas obtained are valid for n equal to one, two and three.

We prove that if $M \longrightarrow P$ is a small cover of a compact right-angled hyperbolic polyhedron $P$ then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient. As a corollary, if $\pi_1(M)$ is commensurable…

We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduces a notion of the secondary strength of a polynomial, and…

We show that the sum of ranks of two matrix polynomials is the same as the sum of the rank of the matrix obtained by applying the greatest common divisor of the polynomials, with the rank of the matrix obtained by applying the lowest common…

If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a…

We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector…

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…

If two homogeneous IFSs satisfying the OSC with opposite common contraction factors share the same attractor on the real line, we show that this attractor is symmetric. This answers a question of Feng and Wang [Adv. Math. 222 (2009),…

The symmetric function theorem states that a polynomial that is invariant under permutation of variables, is a polynomial in the elementary symmetric polynomials. We deduce this classical result, in the analytic setting, from the…