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We bound the Pythagoras number of a real projective subvariety: the smallestpositive integer $r$ such that every sum of squares of linear forms in itshomogeneous coordinate ring is a sum of at most $r$ squares. Enhancing existingmethods, we exhibit three distinct upper bounds involving known invariants. Incontrast, our lower bound depends on a new invariant of a projective subvarietycalled the quadratic persistence. Defined by projecting away from points, thisnumerical invariant is closely related to the linear syzygies of the variety.In addition, we classify the projective subvarieties of maximal andalmost-maximal quadratic persistence, and determine their Pythagoras numbers.

Sums of Squares and Quadratic Persistence on Real Projective Varieties