Basic Properties of Real Analytic and Semianalytic Germs
On the Real Spectrum of a Ring of Global Analytic Functions
On Hilbert's 17th Problem and Real Nullstellensatz for Global Analytic Functions.
On the connected components of a global semianalytic set.
Central Orderings in Fields of Real Meromorphic Function Germs.
Semialgebraic and semianalytic sets
On the Homology of Metacyclic Coverings.
On the Pythagoras numbers of real analytic set germs
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 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.
On global Nash functions
Separation, factorization and finite sheaves on Nash manifolds
On the finiteness of Pythagoras numbers of real meromorphic functions
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...
On the pythagoras numbers of real analytic surfaces
Global problems on Nash functions.
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
We study here several finiteness problems concerning affine Nash manifolds and Nash subsets . Three main results are: (i) A Nash function on a semialgebraic subset of has a Nash extension to an open semialgebraic neighborhood of in , (ii) A Nash set that has only normal crossings in can be covered by finitely many open semialgebraic sets equipped with Nash diffeomorphisms such that , (iii) Every affine Nash manifold with corners is a closed subset of an affine Nash manifold...
Page 1