Positive Polynomials on closed boxes

TL;DR

This paper proves that positive polynomials on closed boxes in two and higher dimensions can be expressed as Bernstein polynomials with strictly positive coefficients, providing two different proof strategies and extending to higher dimensions.

Contribution

It introduces two novel proofs for representing positive polynomials on closed boxes as Bernstein polynomials with positive coefficients, and extends the results to higher dimensions.

Findings

Positive polynomials on closed boxes can be expressed as Bernstein polynomials with positive coefficients.

Two different proof strategies are provided for the two-dimensional case.

The results are extended to polynomials in higher dimensions.

Abstract

We present two different proofs that positive polynomials on closed boxes of $\mathbb{R}^2$ can be written as bivariate Bernstein polynomials with strictly positive coefficients. Both strategies can be extended to prove the analogous result for polynomials that are positive on closed boxes of $\mathbb{R}^n$, $n>2$.