RecenzjaEffective Methods in Algebraic Geometry

ROCZNIKI POLSKIEGO TOWARZYSTWA M ATEM ATY CZNEGO Séria III: MATEMATYKA STOSOWANA XXXVI (1993)

Recenzja

Effective Methods in Algebraic Geometry Edited by T. Mora— (Universita di Genowa) (Italy)

C. Traverso— (Universita di Pisa) (Italy) Progress in Mathematics, vol. 94

Birkhâuser Verlag A.G. 1991, ISBN 3-7643-3546-7

This book is the proceeding of the symposium “MEGA-90— Effective Methods in Algebraic Geometry” , held in Castiglioncello, Italy, in April 17-21, 1990. The main subject of the articles can be summarized as follows:

Algorithms in algebra and (semi-) algebraic geometry, their effectiveness and complexity.

We reproduce below the Table of Contents of the book, since many titles reflect perfectly the main ideas of talks. Then we classify the papers into seven areas, completing the titles by short comments (often drawn out from author’s abstracts), which have to be read only together with titles. To avoid the superfluous repetition we only refer to the first page number of each article.

TABLE OF CONTENS

Forevord: ' v

Addresses of contributors: xi

Riccardo Benedetti: On Lack of Effectiveness in Semi-algebraic Ge-

ometry 1

Edward Bierstone, Pierre Milman: A simple constructive proof of

Canonical Resolution of Singularities 11

Leonardo Caniglia, Jorge A. Guccione, Juan J. Guccione: Local

Membership Problems for Polynomial Ideals 31

Marc Chardin: Un Algorithme pour le Calcul des Résultants 47 Felipe Cucker, Laureano Gonzalez Vega, Francesc Rossello: On al-

gorithms for real algebraic plane curves 63

Alicia M. Dickenstein, Carmen Sessa: Duality methods for the mem-

bership problem 89

130 Recenzja

André Galligo: Exemples d’ensembles de Points en Position Uni-

forme 105

Giovanni Gallo, Bhubaneswar Mishra: Efficient Algorithms and

Bounds for Wu-Ritt Characteristic Sets 119

Tatiana G ate va-Ivanova: Noetherian Properties and Growth of some

Associative Algebras 143

Gerard van der Geer: Codes and Elliptic Curves 159 Marc Giusti, Joos Heintz: Algorithmes - disons rapides - pour la

decomposition d’une variété algébrique en composantes irrédu-

ctibles et équidimensionnelles 169

Dimitri Yu. Grigor’ev: Complexity of Solving Systems of Linear

Equations over the Rings of Differential Operators 195 Teresa Krick, Alessandro Logar: Membership problem, Represen-

tation problem and the Computation of the Radical for one-

dimensional Ideals 203

Lakshman Y.N., Daniel Lazard: On the Complexity of Zero-dimen-

sional Algebraic Systems 217

Lakshman Y.N.: A Single Exponential Bound on the Complexity of

Computing Grôbner Bases of Zero Dimensional Ideals 227 Lars Langemyr: Algorithms for a Multiple Algebraic Extension 235 Henri Lombardi, Marie-Françoise Roy: Elementary constructive the-

ory of ordered fields 249

Henri Lombardi: Effective real Nidlstellensatz and variants 263 Gennadi I. Malashonok: Algorithms for the Solution of Systems of

Linear Equations in Commutative Rings 289

Roger Marlin: Une conjecture sur les anneaux de Chow A(G ,Z) ren-

forcée par un calcul formel 299

Jean-François Mestre: Construction de courbes de genre 2 à partir

de leurs modules 313

H. Michael Môller: Computing Syzygies à la Gaufi-Jordan 335 Jose L. Montana, Luis M. Pardo, Tomas Recio: The non-scalar

Model of Complexity in Computational Geometry 347 Bernard Mourrain: Géométrie et Interpretations Génériques, un Al-

gorithme 363

François Ollivier: Canonical Bases: Relations with Standard Bases, Finiteness Conditions and Application to Tame Auto-

morphisms 379

Gerhard Phster: The tangent cone algorithm and some applications

to local algebraic geometry 401

Jean-François Pommaret, Akli Haddak: Effective Methods for Sys-

tems of Algebraic Partial Differential Equations 411

Daniel Richardson: Finding roots of equations involving functions

defined by first order algebraic differential equations 427 Fabio Rossi, Walter Spangher: Some Effective Methods in the Open-

ness of Loci for Cohen-Macaulay and Gorenstein Properties 441 Marie-Françoise Roy, Aviva Szpirglas: Sign determination on zero

dimensional sets 457

Kiyoshi Shirayanagi: A Classification of Finite-dimensional Mono-

mial Algebras 469

Elisabetta Strickland: An algorithm related to compactifications of

adjoint groups 483

Nikolaj N. Vorobjov (Jr.): Deciding Consistency of Systems of Poly-

nominal in Exponent Inequalities in Subexponential Time 491

Recenzja 131

A S S O C IA T IV E A L G E B R A

143. Let A be a graded standard finitely presented ordered algebra, then A is right noetherian iff A has polynomial growth.

469. Algebras are classified by means of trees of monomials, related to some posets.

C O M M U T A T IV E A L G E B R A , M E M B E R S H IP P R O B L E M ((MP for short): decide whether a given polynomial belongs to a given ideal, or a snb-ring)

31. A generalized MP.

89. The MP involves algorithms of considerably lower complexity, if the underlying algebraic varieties are of special kind.

203. Complexity of MP. Elementary approach for computing radicals.

379. Convenient generators of ideals and sub-rings in polynomial rings k[n].

Testing a conjecture on automorphisms of k[n].

C O M M U T A T IV E A L G E B R A , O T H E R

47. A method of calculating the resultant of three polynomials.

119. Characteristic sets (CS) for polynomial ideals have applications in Computerized Algebra, Automatic Theorem Proving and Computer Vision. Optimal algorithms for finding CS are presented.

217. A probabilistic algorithm for solving a polynomial system of equations (provided there are finitely many solutions).

227. The primary decomposition theory for ideals is involved.

235. Computing products and inverse in fields of algebraic numbers.

249. Constructive proofs for ordered fields, eliminating the use of the axiom of choice.

263. Algorithm related to Nullstellensatz for ordered fields.

289. Comparison with methods of Dodgson and Bareiss.

132 Recenzj a

SEMI ALGEBRAIC GEOMETRY

1. Some natural functions in semi algebraic geometry cannot be upper bounded by any computable function.

347. Computing complexity in semi algebraic geometry.

457. Complexity of deciding the membership of the solutions of some trian- gular systems, in semi algebraic sets.

491. A generalization of an algorithm checking if a system of polynomial inequalities is not contradictory.

ALGEBRAIC GEOMETRY, CURVES

63. New algorithms for computations of the topology of real algebraic plane curves.

313. Algorithm for finding equations of curves corresponding to given points of the moduli space.

ALGEBRAIC GEOMETRY, OTHER 11. An explicit algorithm for desingularizations.

106. Configuration of points in an affine space, and Hilbert polynominal.

159. Codes correspond to families of algebraic curves over finite fields. Al- gebraic geometry is applied to the codes theory and vice versa.

169. The paralelization of the “title” algorithms is discussed. An application to compute the degree and Chow form of a variety is given.

299. Algorithms for computing Chow rings for algebraic groups and homo- geneous spaces.

363. Study of geometric configurations of points by decomposing families of such configurations into irreducible components.

401. Producing a counter example to a conjectured extention of a theorem on the Milnor number for hypersurfaces, to complete intersections.

441. Computing explicit equations for (the complement of) the set of Cohen- Macaulay and Gorenstein points on varieties.

483. Finding the equivariant cohomology of G-equivariant embeddings;

G an algebraic group.

DIFFERENTIAL EQUATIONS

195. An estimation of complexity of solving systems of linear equations over a Weyl algebra.

411. The author claims: “The formal theory of PDE is just the missing tool for studing linear and nonlinear systems of PDE by means of computer algebra” .

427. From author’s abstract: “A method is given for approximating solu- tions of f ( x ) = 0, for / in a certain class F of real valued analytic functions of one real variable” .

Jerzy Jurkiewicz, Jan Rempala