Rational certificates of positivity on compact semialgebraic sets

TL;DR

This paper proves that positive polynomials on compact semialgebraic sets with rational coefficients can be represented with sums of squares over Q, extending classical theorems to rational certificates.

Contribution

It extends Schm"udgen's and Putinar's theorems by showing rational certificates of positivity exist under certain conditions, including the use of a polynomial N - sum X_i^2.

Findings

Rational certificates of positivity are possible for compact semialgebraic sets.

Representation of positive polynomials can be achieved over Q.

Includes a polynomial N - sum X_i^2 in the generators for Putinar's theorem.

Abstract

Schm\"udgen's Theorem says that if a basic closed semialgebraic set K = {g_1 \geq 0, ..., g_s \geq 0} in R^n is compact, then any polynomial f which is strictly positive on K is in the preordering generated by the g_i's. Putinar's Theorem says that under a condition stronger than compactness, any f which is strictly positive on K is in the quadratic module generated by the g_i's. In this note we show that if the g_i's and the f have rational coefficients, then there is a representation of f in the preordering with sums of squares of polynomials over Q. We show that the same is true for Putinar's Theorem as long as we include among the generators a polynomial N - \sum X_i^2, N a natural number.