Jesús M. Ruiz

The basic theory of power series

I Power Series.- 1 Series of Real and Complex Numbers.- 2 Power Series.- 3 Ruckerts and Weierstrasss Theorems.- II Analytic Rings and Formal Rings.- 1 Mathers Preparation Theorem.- 2 Noethers Projection Lemma.- 3 Abhyankars and Ruckerts Parametrization.- 4 Nagatas Jacobian Criteria.- 5 Complexification.- III Normalization.- 1 Integral Closures.- 2 Normalization.- 3 Multiplicity in Dimension 1.- 4 Newton-Puiseuxs Theorem.- IV Nullstellensatze.- 1 Zero Sets and Zero Ideals.- 2 Ruckerts Complex Nullstellensatz.- 3 The Homomorphism Theorem.- 4 Rislers Real Nullstellensatz.- 5 Hilberts 17th Problem.- V Approximation Theory.- 1 Tougerons Implicit Functions Theorem.- 2 Equivalence of Power Series.- 3 M. Artins Approximation Theorem.- 4 Formal Completion of Analytic Rings.- 5 Nash Rings.- VI Local Algebraic Rings.- 1 Local Algebraic Rings.- 2 Chevalleys Theorem.- 3 Zariskis Main Theorem.- 4 Normalization and Completion.- 5 Efroymsons Theorem.- Bibliographical Note.

Sums of two squares in analytic rings

Abstract. We study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions. This is a basic useful property of the plane, but difficult to check in other cases; in particular, what about

Some remarks on pythagorean real curve germs

z^2=xy

On orderings in real surfaces

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Central Orderings in Fields of Real Meromorphic Function Germs.

z^2=yx^2-y^3

Sums of squares in real rings

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ON THE PYTHAGORAS NUMBERS OF REAL ANALYTIC SURFACES

z^2=x^3+y^4

The New Market effect on return and volatility of Spanish stock indexes

or

On rings of semialgebraic functions

z^2=x^3-xy^3

Disentangling the Effects of Supply and Demand Shocks: A DSGE Model for the Spanish Economy☆

? In fact, the unique positive examples we can find are the Brieskorn singularity, the union of two planes in 3-space and the Whitney umbrella. Conversely, we prove that a complete intersection with that property (other than the seven embedded surfaces already mentioned) must be a very simple deformation of the two latter, namely, \[ z^2=x^2+(-1)^ky^k,k\ge3,\quad\mbox{or}\quad z^2=yx^2+(-1)^ky^k,k\ge4. \] In particular, except for the stems