Sums of squares on the hypercube

on direct sums of baer modules

the notion of baer modules was defined recently

On Sums of Three Squares

(1) r3(n) = 4πn S3(n), where the singular series S3(n) is given by (16) with Q = ∞. While in principle this exact formula can be used to answer almost any question concerning r3(n), the ensuing calculations can be tricky because of the slow convergence of the singular series S3(n). Thus, one often sidesteps (1) and attacks problems involving r3(n) directly. For example, concerning the mean valu...

The Hurwitz Theorem on Sums of Squares

This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...

Sums of distinct squares

Sums-of-Squares Formulas

The following is the extended version of my notes from my ATC talk given on June 4, 2014 at UCLA. I begin with a basic introduction to sums-of-squares formulas, and move on to giving motivation for studying these formulas and discussing some results about them over the reals. More recent techniques have made it possible to obtain similar results over arbitrary fields, and some of these are disc...