##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Frank Sottile

#### Clay Mathematical Institute & MSRI

## Certificates of algebraic positivity

##### Abstract:

Positivity is a distinguishing property of the field of real numbers. Writing a polynomial as a sum of squares gives a certificate that it is positive. Hilbert showed that a positive homogeneous quartic polynomial in three variables (ternary quartic) is a sum of three squares of quadratic polynomials. He also showed that there are positive polynomials of every higher degree or greater number of variables with no such sum of squares representations. This led to his 17th problem - to determine whether a positive polynomial is a sum of squares of rational functions. This was answered in the affirmative by Artin in 1926. Recently, positive polynomials have have undergone a revival. In the 1990s Lasserre realized that recent theoretical results from real algebraic geometry and semi-definite programming could be combined to give effective algorithms for solving a class of relaxations of hard optimization problems. The relaxation replaces positivity by sum of squares representation. I will briefly survey the history of positive polynomials and these modern applications, and then discuss a recent strengthening of Hilberts Theorem on ternary quartics: a positive ternary quartic is a sum of squares in exactly 8 inequivalent ways.

Host:

### April 8, 2004

### 4:00 PM

### AP&M 6438

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