Sums of squares on hypersurfaces

TL;DR

This paper investigates the Pythagoras number of certain rings involving square roots of polynomials, establishing that it is infinite under mild conditions on the polynomial.

Contribution

It proves that the Pythagoras number of rings of the form R[x,y,√f(x,y)] is infinite for a broad class of polynomials, extending understanding of sums of squares in these rings.

Findings

Pythagoras number is infinite for rings R[x,y,√f(x,y)] under mild conditions.

The result applies to a wide class of polynomials f(x,y).

Advances the theory of sums of squares on hypersurfaces.

Abstract

We show that the Pythagoras number of rings of type $\mathbb{R}[x,y, \sqrt{f(x,y)}]$ is infinite, provided that the polynomial $f(x,y)$ satisfies some mild conditions.