##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Greg Blekherman

#### Virginia Tech

## Nonnegative Polynomials and Sums of Squares: Real Algebra meets Convex Geometry

##### Abstract:

A multivariate real polynomial is non-negative if its value is at least zero for all points in $\mathbb{R}^n$. Obvious examples of non-negative polynomials are squares and sums of squares. What is the relationship between non-negative polynomials and sums of squares? I will review the history of this question, beginning with Hilbert's groundbreaking paper and Hilbert's 17th problem. I will discuss why this question is still relevant today, for computational reasons, among others. I will then discuss my own research which looks at this problem from the point of view of convex geometry. I will show how to prove that there exist non-negative polynomials that are not sums of squares via ``naive" dimension counting. I will discuss the quantitative relationship between non-negative polynomials and sums of squares and also show that there exist convex polynomials that are not sums of squares.

Hosts: Bill Helton and Jiawang Nie

### October 29, 2009

### 4:00 PM

### AP&M 6402

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