# Exponential convexifying of polynomials

## TL;DR

This paper proves that multiplying a positive polynomial by an exponential function with a sufficiently large exponent makes it strongly convex on convex semialgebraic sets, aiding in finding polynomial critical points.

## Contribution

It introduces a method to convexify polynomials via exponential transformations, extending to unbounded sets under certain positivity conditions.

## Key findings

- Existence of an exponent N for strong convexity on bounded sets.
- Extension to unbounded sets with additional positivity assumptions.
- Application to critical point search of polynomials.

## Abstract

Let $X\subset\mathbb{R}^n$ be a convex closed and semialgebraic set and let $f$ be a polynomial positive on $X$. We prove that there exists an exponent $N\geq 1$, such that for any $\xi\in\mathbb{R}^n$ the function $\varphi_N(x)=e^{N|x-\xi|^2}f(x)$ is strongly convex on $X$. When $X$ is unbounded we have to assume also that the leading form of $f$ is positive in $\mathbb{R}^n\setminus\{0\}$. We obtain strong convexity of $\varPhi_N(x)=e^{e^{N|x|^2}}f(x)$ on possibly unbounded $X$, provided $N$ is sufficiently large, assuming only that $f$ is positive on $X$. We apply these results for searching critical points of polynomials on convex closed semialgebraic sets.