An elementary recursive bound for effective Positivstellensatz and Hilbert 17-th problem

TL;DR

This paper establishes elementary recursive degree bounds for Positivstellensatz and Hilbert 17-th problem, providing explicit tower-of-exponentials bounds based on polynomial parameters.

Contribution

It introduces a new elementary recursive bound for expressing nonnegative polynomials as sums of squares of rational functions, advancing the understanding of these algebraic problems.

Findings

Derived a tower of five exponentials as degree bounds

Provided explicit bounds depending on polynomial number, degree, and variables

Enhanced the theoretical understanding of Positivstellensatz and Hilbert 17-th problem

Abstract

We prove elementary recursive bounds in the degrees for Positivstellensatz and Hilbert 17-th problem, which is the expression of a nonnegative polynomial as a sum of squares of rational functions. We obtain a tower of five exponentials. A precise bound in terms of the number and degree of the polynomials and their number of variables is provided in the paper.