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
Official URL: http://www.comptesrendus.math.ca/
"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
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”)."
|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|
|Deposited On:||18 Apr 2012 08:59|
|Last Modified:||18 Apr 2012 08:59|
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