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Contemporary Mathematics A New Look at Sums of Squares

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 two-square, four-square, and three-square
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


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics