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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 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 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:14 Jul 2014 11:36

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