Sums of Squares on Hypersurfaces

We show that the Pythagoras number of rings of type R[x,y,f(x,y)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}[x,y, \sqrt{f(x,y)}]$$\end{document} is infinite, provided that the polynomial f(x, y) satisfies some mild conditions.