Martí Salat | Searcharxiv

arXiv2018

On the Coefficients of the Permanent and the Determinant of a Circulant Matrix. Applications

Let $d(N )$ (resp. $p(N )$) be the number of summands in the determinant (resp. permanent) of an $N\times N$ circulant matrix $A = (a_{ij} )$ given by $a_{ij} = X_{i+j}$ where $i + j$ should be considered $\mod N$ . This short note is devoted to prove that $d(N ) = p(N )$ if and only if $N$ is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property.

math.AG↗

arXiv2017

On the classification of Togliatti systems

In [MeMR], Mezzetti and Miró-Roig proved that the minimal number of generators $μ(I)$ of a minimal (smooth) monomial Togliatti system $I\subset k[x_{0},\dotsc,x_{n}]$ satisfies $2n+1\le μ(I)\le \binom{n+d-1}{n-1}$ and they classify all smooth minimal monomial Togliatti systems $I\subset k[x_{0},\dotsc,x_{n}]$ with $2n+1\le μ(I)\le 2n+2$. In this paper, we address the first open case. We classify all smooth monomial Togliatti systems $I\subset k[x_{0},\dotsc,x_{n}]$ of forms of degree $d\ge 4$ with $μ(I)=2n+3$ and $n\ge 2$ and all monomial Togliatti systems $I\subset k[x_0,x_1,x_2]$ of forms of degree $d\ge 6$ with $μ(I)=7$.

math.AG↗