Sums of squares of regular functions on rational surfaces

2024/07/29 by Kowalczyk, Tomasz
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.20378

Abstract

We study the sums of squares on cylinders of the form X × \mathbbA_K for a (weakly) factorial curve C. We prove the equality of the Pythagoras numbers of the ring of regular functions on the cylinder with that of the field of rational functions. We then apply these results to the case of (uniformly) rational varieties. We show that if X is a nonsingular rational algebraic surface over the reals, then the Pythagoras number of the ring of regular functions on X is bounded above by 12.

Related

• Polynomials nonnegative on the cylinder
• Sums of squares on reducible real curves
• The Pythagoras number of a rational function field in two variables
• Uniform Bounds for the Number of Rational Points on Symmetric Squares of Curves with Low Mordell-Weil Rank
• The Pythagoras number of fields of transcendence degree 1 over ℚ