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Description: |
Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). We are interested in the existence of rational function representations of small degree. We will show how to derive a general upper bound in terms of the Hilbert function of X, and apply it to the hypercube C={0,1}^n, where we can use the representation theory of S_n to prove tightness. Using this, we will construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree.
Joint work with Grigoriy Blekherman and James Pfeiffer.
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