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A sufficient condition for a polynomial to be sum of 2mth powers of rational functions

Bradley Delso, Margarita (1989) A sufficient condition for a polynomial to be sum of 2mth powers of rational functions. Comptes Rendus Mathématiques de l'Académie des Sciences. Mathematical Reports of the Academy of Science , 11 (2). pp. 63-68. ISSN 0706-1994

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Abstract

"The paper deals with representation of polynomials in several variables over a real closed field as
sums of 2mth powers of rational functions. It has been proved by A. Prestel [M´em. Soc. Math.
France (N.S.) No. 16 (1984), 53–65; MR0792493 (87c:12002)] that even in one variable over R,
the set of polynomials of given degree d which are sums of a given number of 2mth powers of
rational functions is not semialgebraic in the space of coefficients Rd+1 (there is no bound for the
degree of the rational functions involved). On the other hand, E. Becker [J. Reine Angew. Math.
307/308 (1979), 8–30; MR0534211 (80k:12034)] has given valuative necessary and sufficient
conditions for an element of a field to be sum of 2mth powers.
In this paper, the author establishes a criterion which, when satisfied by a polynomial f, implies
that f also agree with Becker’s criterion and then that f is a sum of 2mth powers of rational
functions. The advantage is that this criterion has some semialgebraic nature and can be used
to show that a certain class of semialgebraic sets of polynomials in several variables over a real
closed field are sets of sums of a bounded number of 2mth powers of rational functions of bounded
degree.
The reading is sometimes made difficult by the style, some typographic irregularities (for example,
one has to make the assignments V := v, k := K) and the use of notations which are not
previously defined (like “supporting hyperplane” or “U1/N”)."

Item Type:Article
Uncontrolled Keywords:real fields; positive definite; semialgebraic sets; elementary properties; sum of 2m-th powers of rational functions; stable sets
Subjects:Sciences > Mathematics > Logic, Symbolic and mathematical
ID Code:14816
Deposited On:18 Apr 2012 08:59
Last Modified:18 Apr 2012 08:59

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