A sufficient condition for a polynomial to be sum of 2mth powers of rational functions

Abstract

"The paper deals with representation of polynomials in several variables over a real closed field as sums of 2mth powers of rational functions. It has been proved by A. Prestel that even in one variable over R, the set of polynomials of given degree d which are sums of a given number of 2mth powers of rational functions is not semialgebraic in the space of coefficients Rd+1 (there is no bound for the degree of the rational functions involved). On the other hand, E. Becker has given valuative necessary and sufficient conditions for an element of a field to be sum of 2mth powers. In this paper, the author establishes a criterion which, when satisfied by a polynomial f, implies that f also agree with Becker’s criterion and then that f is a sum of 2mth powers of rational functions. The advantage is that this criterion has some semialgebraic nature and can be used to show that a certain class of semialgebraic sets of polynomials in several variables over a real closed field are sets of sums of a bounded number of 2mth powers of rational functions of bounded degree. The reading is sometimes made difficult by the style, some typographic irregularities (for example, one has to make the assignments V := v, k := K) and the use of notations which are not previously defined (like “supporting hyperplane” or “U1/N”)."