Justin Chen : Sums of squares, Hankel index, and almost real rank

Posted by Wenliang Zhang , part of the Commutative Algebra Seminar.

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Nov. 10, 2021, 3 p.m.

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Abstract

The difference between nonnegative polynomials and sums of squares is an important topic in real algebraic geometry, and the Hankel index of a variety is a natural (but subtle) invariant that quantifies this difference. Surprising connections were found between the Hankel index and the commutative algebra of the variety, namely the N_{2,p} property of linear syzygies for the free resolution - although this only provides half the story. For curves of almost minimal degree, we complete the picture, by determining the Hankel index in terms of a new rank called almost real rank, which interpolates between real (Waring) rank and complex border rank. This is joint work with Greg Blekherman and Jaewoo Jung.