JournalsjemsVol. 24, No. 3pp. 925–965

Sums of squares and quadratic persistence on real projective varieties

  • Grigoriy Blekherman

    Georgia Tech, Atlanta, USA
  • Rainer Sinn

    Universität Leipzig, Germany
  • Gregory G. Smith

    Queen's University, Kingston, Canada
  • Mauricio Velasco

    Universidad de los Andes, Bogotá, Colombia
Sums of squares and quadratic persistence on real projective varieties cover
Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

We bound the Pythagoras number of a real projective subvariety: the smallest positive integer rr such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of at most rr 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.

Cite this article

Grigoriy Blekherman, Rainer Sinn, Gregory G. Smith, Mauricio Velasco, Sums of squares and quadratic persistence on real projective varieties. J. Eur. Math. Soc. 24 (2022), no. 3, pp. 925–965

DOI 10.4171/JEMS/1108