On the minimum of a polynomial function on a basic closed semialgebraic set and applications

TL;DR

This paper provides explicit bounds on the algebraic degree and minimum value of polynomial functions on semialgebraic sets, with applications to component separation in real algebraic geometry.

Contribution

It introduces explicit bounds for the algebraic degree and minimum value of polynomials on semialgebraic sets, advancing quantitative real algebraic geometry.

Findings

Explicit upper bounds for algebraic degree of minima.

Explicit lower bounds for the absolute value of minima.

Lower bounds for separation of disjoint semialgebraic components.

Abstract

We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is not zero. As an application, we obtain a lower bound for the separation of two disjoint connected components of basic closed semialgebraic sets, when at least one of them is compact.