Certificates of non-negativity on semi-algebraic subsets of cylinders

GABRIELA JERONIMO; DANIEL PERRUCCI

Eindhoven

Conferencia; SIAM Conference on Applied Algebraic Geometry; 2023

Socitey for Industrial and Applied Mathematics

A certificate of non-negativity (resp. positivity) for a polynomial $f\in \mathbb{R}[\mathbf{x}]= \mathbb{R}[x_1, \dots, x_n]$ on a semialgebraic set $S\subset \mathbb{R}^n$ is an algebraic identity that makes evident the fact that $f(x) \ge 0$ (resp. $f(x)>0$) for every $x\in S$. These certificates go back to the classical Positivstellenstaz proved by Krivine in 1964, and they have been widely studied and applied since then.In this talk, we will present a new certificate of non-negativity for a polynomial $f$ that is positive on a non-compact semialgebraic set $S=\{ x\in \mathbb{R}^n \mid g_1(x)\ge 0, \dots, g_s(x) \ge 0\}$ included in a cylinder. Under certain assumptions on $f$ and the polynomials $g_1, \dots, g_s\in \mathbb{R}[\mathbf{x}]$ defining $S$, we will show the existence of the proposed certificate of non-negativity of $f$ and explain how one can obtain an upper bound for the degrees of the polynomials appearing in the representation.