Deciding non-negativity of generic polynomials based on Sturm’s theorem

Date: 14:00-15:00, 02 October 2025

Venue: Room 301, A5, Institute of Mathematics

Abstract: We address the problem of deciding the non-negativity of a real polynomial $f$. To this end, we employ the critical value polynomial $varphi_f,$ whose roots are the complex critical values of $f$. First, we show that, under a genericity assumption, this decision problem reduces to determining whether $varphi_f$ has only non-negative roots. This reduction allows us to apply Sturm’s theorem to resolve the problem. Second, we propose a symbolic algorithm for deciding non-negativity and analyze its bit complexity in the case where the input polynomial has rational coefficients. The talk is based on joint work with Nguyen Hong Duc and Akiko Takeda