Positivity of continuous piecewise polynomials

TL;DR

This paper extends positivity certificates for polynomials to continuous piecewise polynomials on simplicial complexes, providing explicit bounds and covering non-negative cases in one dimension.

Contribution

It applies Putinar's theorem to piecewise polynomials on simplicial complexes and improves results for 1D non-negative polynomials with explicit degree bounds.

Findings

Positivity certificates for piecewise polynomials on simplicial complexes.

Extension of Putinar's theorem to this setting.

Explicit degree bounds for 1D non-negative piecewise polynomials.

Abstract

Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for strictly positive polynomials on compact sets can be applied in the case of strictly positive piecewise polynomials on a simplicial complex. In the 1-dimensional case, we improve this result to cover all non-negative piecewise polynomials and give explicit degree bounds.