Representation of a real polynomial f(X) as a sum of 2mth powers of rational functions.

Abstract

"In this paper the authors discuss the representation of a polynomial as a sum of 2mth powers of rational functions. Of course, it is known that the number of rational functions involved can be bounded in terms of m, but also that the degrees of the denominators depend heavily on the coefficients of the given polynomial [Prestel, M´em. Soc. Math. France (N.S.) No. 16 (1984), 53– 65; MR0792493 (87c:12002)]. In order to understand that dependence for polynomials over the real numbers, the authors prove Theorem A. There exists a computable function such that, for all monic polynomials f of degree d such that (i) 2m | d, (ii) kfk < N, and (iii) every monic polynomial g with kP f − gk < 1/M is strictly positive definite, there is a representation f = s i=1 g2m i /h2m, with degree(h) (d,N,M) (the norm of a polynomial is the largest absolute value of its coefficients). We stress that this result is stated over the real numbers, but the proof involves passing to all real closed fields, applying there Becker’s valuative criterion for sums of 2mth powers and using the compactness theorem to get the bound . Despite this fact, the restrictions in the statement make it really interesting over the reals. Indeed, as the authors remark, any positive semidefinite polynomial can be written in the form a(X − 1)d(1) · · · (X − r)d(r)f, where a is positive, the i are the (different) real roots with multiplicities d(i), and f is strictly positive definite. Thus to represent this product as a sum of 2mth powers, one checks first that 2m divides all the d(i)’s, and then expresses f as such a sum. But now Theorem A applies to f for suitable N and M only if we are dealing with the real numbers. The authors also point out that the big issue here is to produce the roots of our starting polynomial, and even to show that with them as data, condition (iii) in Theorem A can be made more explicit."