Representation of a real polynomial f(X) as a sum of 2m-th powers of rational functions

Abstract

From Becker’s Satz 2.14 in [B₁] it follows that a polynomial f ∈ ℝ[X] admits a representation

$$ f = \sum\limits_{{i = 1}}^{\sigma } {\frac{{g_i^{{2m}}}}{{{h^{{2m}}}}}} $$

((1))

with gⁱ, h∈ ℝ[X] if and only if f satisfies the following three conditions:

• (i) 2m divides deg f

• (ii) 2m divides the order of every real zero of f

• (iii) f is positive semidefinite

Once f satisfies these conditions, the problem arises how to obtain a representation (1) for f. This paper is concerned with that problem.

The result of this paper was obtained when the first author was working on her thesis [Br] under the supervision of the second author