10 oct 2016 -- 14:30
sala conferenze Tricerri
Abstract.
Given a polynomial p of degree d in n variables $x_1,...,x_n$, there always exists a square matrix $M$ whose entries are affine-linear in the $x_i$ and whose determinant equals $p$. Probably the best-known instance of this is the case where $n=1$ and $M$ is the companion matrix of $p$. Across several areas of mathematics and theoretical computer science the challenge arises to minimise the number $N$ of rows among all such determinantal representations of $p$. I will discuss some of this existing literature, and then zoom in on recent, joint work with Boralevi, van Doornmalen, Hochstenbach, and Plestenjak, in which we require that the entries of $M$, in addition to being affine-linear in the $x_i$, depend polynomially (or even affine-linearly) on the coefficients of $p$. Then $M$ is a uniform determinantal representation of all polynomials of degree at most $d$ in $x_1,...,x_n$. In this setting, which is motivated by Hochstenbach-Plestenjak's work in numerics, we show that the minimal $N$ grows like $C(n) \cdot \sqrt{d^n}$ for $n$ fixed and $d$ tending to infinity.