Sums of squares of linear forms: the quaternions approach

Abstract

Let A = k[y] be the polynomial ring in one single variable y over a field k. We discuss the number of squares needed to represent sums of squares of linear forms with coefficients in the ring A. We use quaternions to obtain bounds when the Pythagoras number of A is ≤ 4. This provides bounds for the Pythagoras number of algebraic curves and algebroid surfaces over k.