Sums of Squares: A Real Projective Story

TL;DR

This paper surveys the historical and recent developments in understanding the relationship between nonnegative polynomials and sums of squares within real algebraic geometry, emphasizing connections to complex algebraic geometry and commutative algebra.

Contribution

It provides a comprehensive overview of the evolution and recent advances in the study of sums of squares on real varieties, revealing unexpected links to other mathematical fields.

Findings

Highlighting recent breakthroughs in sums of squares theory

Connecting real algebraic geometry with complex algebraic geometry

Revealing new algebraic structures related to nonnegative polynomials

Abstract

The relationship between nonnegative polynomials and sums of squares is one of the central questions in real algebraic geometry. A modern approach is to look at nonnegative polynomials and sums of squares on a real variety. We survey the rich history of this area and focus on recent developments highlighting unexpected links to complex algebraic geometry and commutative algebra.