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Book review

Author(s) Eissette, F. (ed.)
Galligo, A. (ed.)
Title Computational Algebraic Geometry
Publisher Birkhäuser
Year of publication 1992
Reviewed by Bazil Pârv

This volume, the 1O9th of the series "Progress in Mathematics", edited by Birkhäuser, gathers together 21 papers on new theoretical and practical aspects of algebraic geometry, presented at MEGA-92 conference (Effective Methods in Algebraic Geometry, Nice, France, April 21-25, 1992). The first MEGA conference was held in Livorno, Italy, on April 17-21, 1990; the 94th issue of the -above mentioned series contains papers presented there.

The main two topics on which the papers are oriented can be: effective methods and complexity issues in commutative algebra, projective geometry, real geometry, algebraic number theory; algebro-geometric methods in algebraic computing and applications. The above mentioned papers are the following:

E. Becker and R. Neuhaus: Computation of real radicals of polynomial ideals

M. Briskin and Y. Yomdin: Semialgebraic geometry of polynomial control problems

M. Chardin: The resultant via a Koszul complex

A.M. Cohen and R.H. Cushman: Gröbner bases and standard monomial theory

C.N. Delzell, L. Gonzalez-Vega, and H. Lombardi: A continuous and rational solution to Hilbert's 17th problem and several cases of the Positivstellensatz

P. Gimenez, M. Morales, and A. Simis: The analytic spread of the ideal of a monomial curve in projective 3-space

D. Grigoriev, M. Karpinski, and M.F. Singer: Computational complexity of sparse real algebraic function interpolation

J.-P. Henry and M. Merle: Shade, shadow and shape

P. Jaworski: Arrangements of singularities and proper partitions of Dynkin diagrams

V.P. Kostov and S.K. Lando: Versal deformations of powers of volume forms

D. Lazard and A. Valibouze: Computing subfields: Reverse of the primitive element problem

A. Lecki and Z. Szafraniec: Applications of Eisenbud-Levine's theorem to real algebraic geometry

S.J. Maybank: Applications of algebraic geometry to computer vision

G. Niesi and L. Robbiano: Disproving Hibi's conjecture with CoCoA or projective curves with bad Hilbert functions

P. Pedersen, M.-F. Roy, and A. Szpirglas: Counting real zeros in the multivariate case

D. Richardson: Finding the number of distinct real roots of sparse polynomials of the form p(x,x^n)

J. Rubio and F. Sergeraet: Locally effective objects and algebraic topology

K. Shirayanagi: Decision of algebra isomorphisms using Gröbner bases

M. Shub and S. Smale: Complexity of Bezout's theorem II: volumes and probabilities

F. Smietanski: A parametrized Nullstellensatz

D. Wang: An ellimination method based on Seidenberg's theory and its applications

The book is typeset; even if it contains several contributions from different authors, the manner in which the text is ranged is unitary and makes the papers pleasant to read.

The volume, whose price/consistency ratio is high enough, is available, in my opinion, only for professional mathematicians and graduate students working in this field.