Publication: A characterization of sums of 2nth powers of global meromorphic functions
Loading...
Official URL
Full text at PDC
Publication Date
1990-08
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
Abstract
Hilbert's 17th Problem asked the following. Let f(x1,⋯,xn) be a real polynomial which for all real values α1,⋯,αn satisfies f(α1,⋯,αn)≥0. Is it true that f=∑(gi/hi)2 for polynomials gi,hi∈R[x1,⋯,xn]? (It was known that f=∑(gi)2, gi∈R[x1,⋯,xn], is not always true.) Artin gave a positive answer to this problem. In this paper the author proves an analogous theorem with f replaced by a real-analytic function on a compact analytic subvariety X of a real analytic manifold, the rational functions gi/hi replaced by meromorphic functions, the power 2 replaced by 2n and the condition "f(α1,⋯,αn)≥0 for all real α1,⋯,αn ''replaced by a suitable condition that reduces in the case n=1, X=Rn, to "f≥0''. This condition is as follows: For every analytic curve σ:(−ε,ε)→X for which (1) the germ of the image of σ at σ(0) is not contained in the germ of the singular set of X at σ(0) and (2) f∘σ(t)=atm+⋯(a≠0), one has a>0 and 2n∣m. The author studies the relationship between curves satisfying (1) and certain valuations on the field of meromorphic functions. On the other hand, Becker's theory of 2nth powers in formally real fields related the question of representing an element as the sum of 2nth powers to the study of valuations on the field. These two theories are combined to yield the 2nth power, analytic analogue of the original Hilbert problem.
Description
UCM subjects
Unesco subjects
Keywords
Citation
E. Becker, Heredirarily Pythagorean fields and orderings of higher level, IMPA Lecture Notes, no. 29, Rio de Janeiro, 1978.
E. Becker, The real holomorphy ring and sums of 2n-th powers, Lecture Notes in Math., vol. 959, Springer-Verlag, Berlin, Heidelberg, and New York, 1982.
L. Bröcker and H.-W. Schülting, Valuations of function fields from the geometrical point of view, J. Reine Angew. Math. 365 (1986), 12-32.
F. Bruhat and H. Whitney, Quelques propriétés fondamentales des ensembles analytiques réels, Comment. Math. Helv. 33 (1959), 132-160.
H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77-99.
A. Grothendieck and J. Dieudonné, Elements de géométrie algebrique, Publ. Math. I.H.E.S. 24 (1965).
J. Frisch, Points de platitude d'un morphisme d'espaces analytiques complexes, Invent. Math. 4 (1967), 118-138.
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79 (1964), 109-326.
H. Hironaka, Introduction to real analytic sets and real analytic maps, Quaderno dei gruppi di ricerca del C.N.R. Ist. Mat. "L. Tonelli", Pisa, 1973.
W. Kucharz, Sums of 2n-th powers of real meromorphic functions (to appear).
F. V. Kuhlmann and A. Prestel, On places of real algebraic function fields, J. Reine Angew. Math. 358 (1984), 181-195.
H. Matsumura, Commutative algebra, 2nd ed., W. A. Benjamin Co., Amsterdam, 1980.
J. Ruiz, On Hilbert's 17th problem and real Nullstellensatz for global analytic functions, Math. Z. 190 (1985), 447-454.
H-W Schülting, Prime divisors on real varieties and valuation theory, J. Algebra 98 (1986), 499-514.
J. C. Tougeron, Idéaux de fonctions différentiables, Ergeb. Math. 71, Springer-Verlag, Berlin, Heidelberg, and New York, 1972.