Estimates of the Pythagoras number of Rm[x1,…,xn]Rm[x1,…,xn] through lattice points and polytopes

Hilbert’s 17th Problem launched a number of inquiries into sum-of-squares representations of polynomials over the real numbers. Choi, Lam, and Reznick gave some bounds on the number of squares required for such a representation and indicated some directions for improving these bounds. In the first part of this paper, we follow their suggestion and obtain some stronger bounds. In the second part, we show that in the case of homogeneous polynomials in three variables, this technique cannot be extended further.