Linear Operators $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ and $K$-Positivity Preserver: A Short Review

TL;DR

This paper reviews recent advances in the theory of linear operators on multivariate polynomial spaces, focusing on $K$-positivity preservers that maintain non-negativity over a subset $K$ of $ ^n$, highlighting key developments and open questions.

Contribution

It provides a concise overview of the latest theoretical progress on $K$-positivity preservers for linear operators on multivariate polynomial spaces.

Findings

Summarizes recent theoretical results on $K$-positivity preservers.

Highlights open problems and future research directions.

Connects operator theory with polynomial positivity over subsets of $ ^n$.

Abstract

In the current short review we present the latest developments on linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$, especially of $K$-positivity preserver, i.e., $Tp\geq 0$ on $K\subseteq\mathbb{R}^n$ for all $p\in\mathbb{R}[x_1,\dots,x_n]$ with $p\geq 0$ on $K$.