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

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1989

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Kluwer Acad. Publ

##### 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."

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Ordered algebraic structures: proceedings of the Caribbean Mathematics Foundation Conference on Ordered Algebraic Structures, Curaçao, August 1988