%0 Book Section
%T Representation of a real polynomial f(X) as a sum of 2mth powers of rational functions.
publisher Kluwer Acad. Publ
%D 1989
%U 0-7923-0489-6
%@ https://hdl.handle.net/20.500.14352/60541
%X "In this paper the authors discuss the representation of a polynomial as a sum of 2mth powersof rational functions. Of course, it is known that the number of rational functions involved canbe bounded in terms of m, but also that the degrees of the denominators depend heavily on thecoefficients 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 thereal numbers, the authors prove Theorem A. There exists a computable function   such that, forall monic polynomials f of degree d such that (i) 2m | d, (ii) kfk < N, and (iii) every monicpolynomial g with kP f − gk < 1/M is strictly positive definite, there is a representation f = si=1 g2mi /h2m, with degree(h)    (d,N,M) (the norm of a polynomial is the largest absolutevalue of its coefficients).We stress that this result is stated over the real numbers, but the proof involves passing to allreal closed fields, applying there Becker’s valuative criterion for sums of 2mth powers and usingthe compactness theorem to get the bound  . Despite this fact, the restrictions in the statementmake it really interesting over the reals. Indeed, as the authors remark, any positive semidefinitepolynomial 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 torepresent 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 onlyif we are dealing with the real numbers. The authors also point out that the big issue here is toproduce 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."
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