Lectures on Nonnegative Polynomials and Sums of Squares

TL;DR

This paper provides an accessible overview of nonnegative polynomials and sums of squares, emphasizing historical context, recent advances, and connections to algebraic geometry, semidefinite programming, and applications.

Contribution

It introduces recent developments and new connections in the theory of nonnegative polynomials and sums of squares, bridging algebraic geometry and optimization.

Findings

Highlights recent theoretical advances.

Explores connections with algebraic geometry.

Discusses applications to semidefinite programming.

Abstract

These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic geometry and commutative algebra. We briefly discuss the connection to semidefinite programming and applications. We expect the reader to be familiar with algebra and fundamentals of algebraic geometry, on the level of a graduate course, and know basic facts about positive semidefinite matrices.