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On the Pythagoras numbers of real analytic set germs

José F. FernandoJesús M. Ruiz — 2005

Bulletin de la Société Mathématique de France

We show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.

On the finiteness of Pythagoras numbers of real meromorphic functions

Francesca AcquistapaceFabrizio BrogliaJosé F. FernandoJesús M. Ruiz — 2010

Bulletin de la Société Mathématique de France

We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real...

Global problems on Nash functions.

Michel CosteJesús M. RuizMasahiro Shiota — 2004

Revista Matemática Complutense

This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings.

Finiteness problems on Nash manifolds and Nash sets

José F. FernandoJosé Manuel GamboaJesús M. Ruiz — 2014

Journal of the European Mathematical Society

We study here several finiteness problems concerning affine Nash manifolds M and Nash subsets X . Three main results are: (i) A Nash function on a semialgebraic subset Z of M has a Nash extension to an open semialgebraic neighborhood of Z in M , (ii) A Nash set X that has only normal crossings in M can be covered by finitely many open semialgebraic sets U equipped with Nash diffeomorphisms ( u 1 , , u m ) : U m such that U X = { u 1 u r = 0 } , (iii) Every affine Nash manifold with corners N is a closed subset of an affine Nash manifold...

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