On the Representation in Sums of Squares for Definite Functions in One Variable Over an Algebraic Number Field

Abstract

1. If R is a real closed field, then Artin [1] showed that every positive definite function f(Xl, . . ., Xj) in the rational function field R (X1,.. ., X.,) may be expressed as a sum of squares, thus solving a problem of Hilbert. However, Artin did not furnish the number of necessary squares. It was Pfister [8] who proved that 2' number of squares would suffice. Whether or not this upper bound is the best possible in general is still an outstanding open question, although for n < 2 this has been settled in the affirmative by Cassels-EllisonPfister [3]. To generalize this Hilbert problem with another ground field K instead of the real closed field R, Pfister asked in [8] whether 2n+2 might serve as an upper bound for K = Q, the rational number field. For n = 0, the theorem of Euler-Lagrange provides the answer. For n =1, Landau [5] showed that every positive definite polynomial in one variable over Q is a sum of eight squares of polynomials. Pourchet, however, recently proved [9] the rather startling fact that five is, in fact, the best possible bound in this case. In this short note we complete Pourchet's work by explicitly determining the best possible bound for K (X) where K is any formally real algebraic number field. We shall call this invariant the height. Thus, the reduced height m(F) of a field F is the least positive integer (or infinity) such that every sum of squares in F is a sum of m(F) number of squares. More precisely, we prove the following: