 
Summary: Contemporary Mathematics
A New Look at Sums of Squares
Larry J. Gerstein
Let R be an integral domain. Among the most basic problems in the theory
of quadratic forms over R is the determination of which nonzero elements in R
can be expressed as a sum of n squares in R, where n is a positive integer. When
R = Z, the classics of this genre are the twosquare, foursquare, and threesquare
theorems of Fermat, Lagrange, and Gauss, respectively (listed here in chronological
order). Of more recent vintage (1964) is this theorem of Cassels: If k is a field of
characteristic not 2, then a polynomial in k[x] is a sum of n squares of polynomials
if and only if it is a sum of n squares of rational functions in k(x). (In slightly more
general form this result is known as the CasselsPfister theorem, a cornerstone of
the algebraic theory of quadratic forms.) My first goal in this paper is to present a
unified approach to all these results. Having done this, I will begin the process of
extending this approach to a wider class of problems on representations by quadratic
forms. At the end of the paper I will raise some questions that arise naturally along
the way.
For a ring R, the symbol n will denote the set of nonzero elements in R that
can be written as a sum of n squares of elements of R.
From now on we will assume R is a principal ideal domain whose quotient field
