From the reviews of the first edition: "It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra . Among the great strengths and most distinctive features is a new, completely unified treatment of the global and local theories. making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike." J.B. Little, MAA, March 2004 The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case.

A Singular Introduction to Commutative Algebra

Commutative ring theory: Regular rings

This paper is devoted to the resolution of zero-dimensional systems in K[X
1, …X

 n
 ], where K is a field of characteristic zero (or strictly positive under some conditions). We follow the definition used in MMM95 and basically due to Kronecker for solving zero-dimensional systems: A system is solved if each root is represented in such way as to allow the performance of any arithmetical operations over the arithmetical expressions of its coordinates. We propose new definitions for solving zero-dimensional systems in this sense by introducing the Univariate Representation of their roots. We show by this way that the solutions of any zero-dimensional system of polynomials can be expressed through a special kind of univariate representation (Rational Univariate Representation):
 where (f,g,g
1, …,g

 n
 ) are polynomials of K[X
1, …, X

 n
 ]. A special feature of our Rational Univariate Representation is that we dont loose geometrical information contained in the initial system. Moreover we propose different efficient algorithms for the computation of the Rational Univariate Representation, and we make a comparison with standard known tools.

Solving Zero-Dimensional Systems Through the Rational Univariate Representation

A criterion for detecting m -regularity

This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

Computational methods in commutative algebra and algebraic geometry

http://www.lix.polytechnique.fr/~giusti/publications/GiLeSa:1999:kro-focm99.pdf

A Gröbner Free Alternative for Polynomial System Solving

Straight--Line Programs in Geometric Elimination Theory

Some Effectivity Problems in Polynomial Ideal Theory

Lower bounds for diophantine approximations

We present a new method for solving symbolically zero-dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straight-line programs with FOR gates. For sequential time complexity measured by the size of these networks we obtain the following result: it is possible to solve any affine or toric zero-dimensional equation system in non-uniform sequential time which is polynomial in the length of the input description and the “geometric degree” of the equation system. Here, the input is thought to be given by a straight-line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). Geometric degree has to be adequately defined. It is always bounded by the algebraic-combinatoric “Bezout number” of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree is much smaller than the Bezout number since it does not take into account multiplicities or degrees of extraneous components (which are at infinity in the affine case or contained in some coordinate hyperplane in the toric case).

Marc Giusti

Papers

A Gröbner Free Alternative for Polynomial System Solving

Straight--Line Programs in Geometric Elimination Theory

Some Effectivity Problems in Polynomial Ideal Theory

Lower bounds for diophantine approximations

When Polynomial Equation Systems Can Be Solved Fast