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

Abstract: The author shows first that the Pythagoras number of 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 of fini­tely generated modules over power series and po­ly­no­mials in two variables. Secondly, he com­pletes the full classifi­ca­tion in 3-space of surface germs on which every 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:


First advisor: M. Coste.

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

Abstract: The author shows the triviality of pairs of 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 that the trivialization is definable too. This requires the use of the definable spectrum in a essential way to solve the lack of integration means. The author proves previously a basic result: an ap­prox­i­mation theorem for 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 the definable triviality off bifurcation sets.

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:


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

Abstract: The author finds a geometric criterion for basicness of semialgebraic sets in algebraic surfaces. This involves se­pa­ration and approximation, and the des­crip­tion of two types of obstructions: local and global. Since these geometric ideas are connected to the algebraic notion of fan, it is essential to know well the theory of fans in a surface, which is achieved by means of generalized Puiseux expansions. Finally, both the geometric and the algebraic views are mixed to produce an algorithm 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:


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

Abstract: The author studies the Py­tha­go­ras number of a real irreducible algebroid curve in 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 into finitely many semialgebraic sets which involves algorithmic (upper and lower) bounds for the Pythagoras number. An ap­plication of this is the determination of Py­tha­gorean curves. Other matters dis­cussed are semicontinuity of the Py­tha­go­ras num­ber, Hilbert's 17th Problem, and change of ground field.

Keywords: Pythagoras number, value semigroup.

Mathematical Subject Classification: 14P15, 32B10.

The main content of this work appears in: