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**PhD's**

(Ph.D. dissertations written under our supervision)

[UCM01] J.F. Fernando:Sums of squares in surface germs.

Abstract:The author shows first that thePythagoras numberof a real analytic surface germ is finite, bounded by a function of its multiplicity and its codimension. This he gets by solving a question concerning Pythagoras numbers offinitely generated modules overpower series and polynomials intwo variables.Secondly, he completes the full classification in 3-space of surface germs on whichevery positive semidefinite analytic germ is a sum of squares of analytic germs(in fact, he shows, of two squares). He also finds examples of not embedded surface germs with that property.

Keywords:Positive semidefinite germ, sum of squares, analytic germ.

Mathematical Subject Classification:14P20, 32S10.To get a PDF file of the thesis (Spanish), click here, for a summary (English) here. The following papers include these results:

J.F. Fernando:On the Pythagoras number of real analytic rings,J. Algebra243(2001) 321-338.J.F. Fernando, J.M. Ruiz:Positive semidefinite germs on the cone,Pacific J. Math.205(2002) 109-118.J.F. Fernando:Positive semidefinite germs in real analytic surfaces,Math. Ann.322(2002) 49-67.

First advisor: M. Coste.

[UCM00] J. Escribano:Definable triviality of families of definable mappings in o-minimal structures.

Abstract:The author shows thetriviality of pairsof proper submersions in any o-minimal structure expanding a real closed field. This means not only that the involved submersions are definable, but most important thatthe trivialization is definabletoo. This requires the use of thedefinable spectrumin a essential way to solve the lack of integration means. The author proves previously a basic result: anapproximation theoremfor definable differentiable maps, which has interest by itself and implies other interesting applications (smoothing of corners,for instance). He finally obtains a nice application concerning thedefinable triviality off bifurcationsets.

Keywords:Triviality of submersions, definable approximation, definable spectrum.

Mathematical Subject Classification:14P20, 03C68, 57R12.To get a PDF file of the thesis (Spanish), click here, for an English version here. These results are in:

J. Escribano:Nash triviality in families of Nash mappings,Ann. Inst. Fourier51(2001) 1209-1228.J. Escribano:Bifurcation sets of definable functions in o-minimal structures,Proc. AMS130(2002) 2419-2424.J. Escribano:Approximation theorems in o-minimal structures,Illinois J. Math.46(2002) 111-128.

[UCM94] P. Vélez:The geometry of fans in dimension 2.

Abstract:The author finds a geometric criterion forbasicnessof semialgebraic setsin algebraic surfaces.This involvesseparationandapproximation,and the description of two types of obstructions: local and global. Since these geometric ideas are connected to the algebraic notion offan,it is essential to know well the theory of fans in a surface, which is achieved by means ofgeneralized Puiseux expansions.Finally, both the geometric and the algebraic views are mixed to produce analgorithm that checks basicness,and exhibits a fan obstruction if there is any.

Keywords:Basic semialgebraic set, fan, separation.

Mathematical Subject Classification:14P10, 13A18.These results appear in:

F. Acquistapace, F. Broglia, P. Vélez:An algorithmic criterion for basicness in dimension 2,Manuscripta Math.85(1995) 45-66.P. Vélez:On fans in real surfaces,J. Pure Appl. Algebra136(1999) 285-296.

[UCM86] J. Ortega:Pythagoras numbers of a real irreducible algebroid curves.

Abstract:The author studies thePythagoras number of a real irreducible algebroid curvein terms of its value semigroup. Fixed a semigroup, he describes explicitely an algebraic set parametrizing all curves with that semigroup of values. Then, the Pythagoras number defines a partition of that set intofinitely many semialgebraic setswhich involvesalgorithmic (upper and lower) boundsfor the Pythagoras number. An application of this is thedetermination of Pythagorean curves.Other matters discussed aresemicontinuity of the Pythagoras number, Hilbert's 17th Problem,andchange of ground field.

Keywords:Pythagoras number, value semigroup.

Mathematical Subject Classification:14P15, 32B10.The main content of this work appears in:

J. Ortega:On the Pythagoras number of a real algebroid irreducible curve,Math. Ann.289(1991) 111-123.