# Optimization and Positivity Certificates of Rational Functions using Bernstein Form

## TL;DR

This paper explores the representation of rational functions in Bernstein form over a simplex, providing bounds for their range, algebraic positivity certificates, and dimension-independent bounds.

## Contribution

It introduces new bounds for rational functions in Bernstein form that do not depend on the dimension, along with algebraic positivity certificates.

## Key findings

- Bounds converge to the true range of the rational function.
- Positivity certificates are derived using algebraic identities.
- Dimension-independent bounds are established.

## Abstract

Rational functions of total degree $l$ in n variables have a representation in the Bernstein form defined over $n$ dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein coefficients over a simplex. Convergence properties of the bounds to the range are reviewed. Algebraic identities certifying the positivity of a given rational function over a simplex are given. Subsequently, a bound established in this work does not depend on the given dimension.