## Experimental Mathematics

- Experiment. Math.
- Volume 21, Issue 4 (2012), 353-361.

### Discriminants, Symmetrized Graph Monomials, and Sums of Squares

Per Alexandersson and Boris Shapiro

#### Abstract

In 1878, motivated by the requirements of the invariant theory of binary forms, J. J. Sylvester constructed, for every graph with possible multiple edges but without loops, its symmetrized graph monomial, which is a polynomial in the vertex labels of the original graph. We pose the question for which graphs this polynomial is nonnegative or a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on the discriminant of the derivative of a univariate polynomial and by an interesting example of P. and A. Lax of a graph with four edges whose symmetrized graph monomial is nonnegative but not a sum of squares.We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.

#### Article information

**Source**

Experiment. Math., Volume 21, Issue 4 (2012), 353-361.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1356038818

**Mathematical Reviews number (MathSciNet)**

MR3004251

**Zentralblatt MATH identifier**

1259.13012

**Subjects**

Primary: 13J30: Real algebra [See also 12D15, 14Pxx] 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 05E05: Symmetric functions and generalizations 05C15: Coloring of graphs and hypergraphs 11R29: Class numbers, class groups, discriminants

**Keywords**

Polynomial sums of squares translation-invariant polynomials graph monomials discriminants symmetric polynomials

#### Citation

Alexandersson, Per; Shapiro, Boris. Discriminants, Symmetrized Graph Monomials, and Sums of Squares. Experiment. Math. 21 (2012), no. 4, 353--361. https://projecteuclid.org/euclid.em/1356038818