# Mathematical Sciences Research Institute

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# Seminar

Fellowship of the Ring, National Seminar: Sums of Squares: From Real to Commutative Algebra January 14, 2021 (01:30 PM PST - 03:00 PM PST)
Parent Program: -- MSRI: Online/Virtual
Speaker(s) Grigoriy Blekherman (Georgia Institute of Technology)
Description

To attend this seminar, you must register in advance, by clicking HERE.

Video

#### Sums of Squares From Real to Commutative Algebra

Abstract/Media

To attend this seminar, you must register in advance, by clicking HERE.

Expository Paper:

https://arxiv.org/abs/2101.05773

A real polynomial is called nonnegative if it takes only nonnegative values. A sum of squares or real polynomials is clearly nonnegative. 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 X, where unexpected links to complex algebraic geometry and commutative algebra appear.

In the first half of the talk I will review the history of the problem, do some examples, and provide a brief overview of the results. Our two guiding questions will be: the relationship between nonnegative polynomials and sums of squares, and the number of squares needed to write any sum of squares on X. I will explain the connection between these questions and properties of the free resolution of the ideal of X: the number of of steps that the resolution only has linear syzygies (property $N_{2,p}$) and the number of steps that linear syzygies persist (the length of the linear strand).

In the second half, I will concentrate on the number of squares, and introduce an invariant of X we call quadratic persistence. Quadratic persistence of X is equal to the least number of points in X such that after projecting from (the span of) these points the ideal of the resulting variety has no quadrics. I will explain how quadratic persistence connects real algebraic geometry and commutative algebra. Joint work with Rainer Sinn, Greg Smith and Mauricio Velasco.

 Notes 8.17 MB application/pdf