# Sums of Squares and Quadratic Persistence on Real Projective Varieties

## TL;DR

This paper introduces new bounds for the Pythagoras number of real projective varieties, utilizing a novel invariant called quadratic persistence, and classifies varieties with maximal quadratic persistence.

## Contribution

It provides the first bounds involving known invariants and introduces quadratic persistence, linking it to syzygies and classifying extremal cases.

## Key findings

- Established three upper bounds for the Pythagoras number.
- Defined and analyzed the quadratic persistence invariant.
- Classified varieties with maximal and almost-maximal quadratic persistence.

## Abstract

We bound the Pythagoras number of a real projective subvariety: the smallest positive integer $r$ such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of at most $r$ squares. Enhancing existing methods, we exhibit three distinct upper bounds involving known invariants. In contrast, our lower bound depends on a new invariant of a projective subvariety called the quadratic persistence. Defined by projecting away from points, this numerical invariant is closely related to the linear syzygies of the variety. In addition, we classify the projective subvarieties of maximal and almost-maximal quadratic persistence, and determine their Pythagoras numbers.