Bradley Delso, Margarita and Prestel, Alexander
(1989)
*Representation of a real polynomial f(X) as a sum of 2mth powers of rational functions.*
In
Ordered algebraic structures : Proceeding of the Caribean mathematics Foundations Conference.
Mathematics and its Applications
(55).
Kluwer Acad. Publ, Dordrecht, pp. 197-207.
ISBN 0-7923-0489-6

## 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."

Item Type: | Book Section |
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Additional Information: | Ordered algebraic structures: proceedings of the Caribbean Mathematics Foundation Conference on Ordered Algebraic Structures, Curaçao, August 1988 |

Subjects: | Sciences > Mathematics > Algebra |

ID Code: | 14815 |

Deposited On: | 18 Apr 2012 09:06 |

Last Modified: | 18 Apr 2012 09:06 |

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