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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: --
Location: 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.

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.

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Sums of Squares From Real to Commutative Algebra