Sums of squares, Hankel index, and almost real rank

Grigoriy Blekherman; Justin Chen; Jaewoo Jung

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arXiv · 2108.06635

Sums of squares, Hankel index, and almost real rank

Grigoriy Blekherman·Justin Chen·Jaewoo Jung

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Abstract

The Hankel index of a real variety $X$ is an invariant that quantifies the difference between nonnegative quadrics and sums of squares on $X$. In [5], the authors proved an intriguing bound on the Hankel index in terms of the Green-Lazarsfeld index, which measures the "linearity" of the minimal free resolution of the ideal of $X$. In all previously known cases this bound was tight. We provide the first class of examples where the bound is not tight; in fact the difference between Hankel index and Green-Lazarsfeld index can be arbitrarily large. Our examples are outer projections of rational normal curves, where we identify the center of projection with a binary form $F$. The Green-Lazarsfeld index of the projected curve is given by the complex Waring border rank of $F$ [15]. We show that the Hankel index is given by the "almost real" rank of $F$, which is a new notion that comes from decomposing $F$ as a sum of powers of almost real forms. We determine the range of possible and typical almost real ranks for binary forms.

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Grigoriy Blekherman, Justin Chen, Jaewoo Jung. 2021-08-15. Sums of squares, Hankel index, and almost real rank. https://arxiv.org/abs/2108.06635

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