Maria Emilia Alonso

A computational model for algebraic power series

Up to now, no computational model is known to perform effective commutative algebra for ideals in a computable ring of formal power series, while the theory for this is quite developed at least since [7]. The particular case of algebraic formal power series comes out naturally, when studying singular points of algebraic varieties, for instance, in the NewtonPuiseux algorithm for determining the analytic branches of a curve at a singular point and, more generally, when studying analytic components of a complex algebraic variety. We propose here to develop a computational model for algebraic formal power series, already introduced in [l], based on a symbolic codification of the series by means of the Implicit Function Theorem, i.e., we will consider algebraic series as the unique solutions of suitable functional equations, which we call Locally Smooth Systems. We then reduce the problem of handling a finite set of algebraic series to some corresponding problem involving suitable polynomial rings. In this model we will show that most of the usual local commutative algebra can be effectively performed on algebraic series, since we can reduce to the polyno-

Local Parametrization of Space Curves at Singular Points

We propose a symbolic computation algorithm for computing local parametrization of analytic branches and real analytic branches of a curve in n-dimensional space, which is defined by implicit polynomial equations. The algorithm can be used in space curve tracing near a singular point, as an alternative to symbolic computations based on resolutions of singularities.

The Big Mother of all Dualities: Möller Algorithm

Abstract Duality was introduced in Computer Algebra in 1982 by Möller and since that has been widely used. We give a survey of Möller algorithm and its applications, presenting a new one to the computation of canonical modules. “Its dual application” allow us to give answer to a question posed to us by Stetter.

An Algorithm on Quasi-Ordinary Polynomials

Let K be a field of characteristic O, and let R denote K[X] or K[[X]]. It is well known that the roots of a polynomial FisinR[Z] are fractional powers series in K[[X 1d/]], where Kmacr is a finite extension of K and disin N, and they can be obtained by applying the Newton Puiseux algorithm. Although this is not true for polynomials in more than one variable, there is an important class of polynomials FisinR[Z] (R=K[[X 1, . . ., X n]]=K[[Xlowbar]]), called quasi-ordinary (QO) polynomials, for which the same property holds (i.e. their roots are fractional power series in Kmacr[[Xlowbar 1d/]]). The goal of the paper is to give an algorithm to compute these fractional power series for K (a computable field) and n=2

On orderings in real surfaces

After describing explicitly all total orderings in the ring R[[x, y]], we prove that each ordering in the quotient field of the ring of germs of real analytic functions at an irreducible point O of a real analytic surface X is defined by a half-branch of the germ at O of some curve on X, which is analytic off the origin. Then follows an analogous result for real algebraic surfaces.

Optimality conditions for a nonconvex set-valued optimization problem

In this paper we study necessary and sufficient optimality conditions for a set-valued optimization problem. Convexity of the multifunction and the domain is not required. A definition of K-approximating multifunction is introduced. This multifunction is the differentiability notion applied to the problem. A characterization of weak minimizers is obtained for invex and generalized K-convexlike multifunctions using the Lagrange multiplier rule.

Computing with algebraic series

Up to now, no computational model is known to perform effective commutative algebra for ideals in a computable ring of formal power series, while the theory for this is quite developped at least since [GAL]. The particular case of algebraic formal power series comes out naturally when studying singular points of algebraic varieties, for instance in Newton-Puiseux algorithm for determining the analytic branches of a curve at a singular point and more generally when studying analytic components of a complex algebraic variety. We propose here to develop a computational model for algebraic formal power series, already introduced in [ALR], based on a symbolic codification of the series by means of the Implicit Function Theorem: i.e. we will consider algebraic series as the gniaue solutions of suitable functional equations. In this model we will show that most of the usual local commutative algebra can be effectively performed on algebraic series. since we can reduce to the polynomial case, where the Tangent Cone Algorithm can be used to effectively perform local algebra. We can give a Tangent Cone Algorithm for ideals in the ring of algebraic formal power series, and so compute standard bases and use an effective version of the method of associated graded rings, to deal with basic local ideal theoretical problems. The main result of our paper is however an effective version of Weierstrass theorems, which allow us to have effective elimination theory for algebraic series and an effective Noether Normalization Lemma. In # 1 we recall without proofs the basic theory of standard bases in rings of formal power series.

Elementary constructive theory of Henselian local rings

2 is devoted to the presentation of the computational model for algebraic series we propose, based on the concept of locallv smooth systems. In

The big Mother of all dualities 2: Macaulay bases

3 we show how to modify a local smooth system to effectively compute with algebraic series. In 8 4 we then give a standard basis algorithm for the ring of algebraic series (and so an effective version of the method of associated graded rings). In 95 we give effective versions of Weierstrass Preparation and Division Theorems, which are used in

Set-relations and optimality conditions in set-valued maps

6 to present algorithms for computing elimination and Noether normal position of an ideal of algebraic formal power series.