Sums of squares, Hankel index, and almost real rank

TL;DR

This paper explores the relationship between the Hankel index and Green-Lazarsfeld index of real varieties, introducing the new concept of almost real rank and providing examples where the previously tight bound is not tight.

Contribution

It introduces the notion of almost real rank for binary forms and demonstrates cases where the bound relating Hankel and Green-Lazarsfeld indices is not tight, expanding understanding of these invariants.

Findings

Hankel index can differ arbitrarily from Green-Lazarsfeld index.

Almost real rank is a new measure related to decomposing forms into almost real powers.

Range of almost real ranks for binary forms is characterized.

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.