A New Criterion for Normal Form Algorithms

In this paper, we present a new approach for computing normal forms in the quotient algebra A of a polynomial ring R by an ideal I. It is based on a criterion, which gives a necessary and sufficient condition for a projection onto a set of polynomials, to be a normal form modulo the ideal I. This criterion does not require any monomial ordering and generalizes the Buchberger criterion of S-polynomials. It leads to a newa lgorithm for constructing the multiplicative structure of a zero-dimensional algebra. Described in terms of intrinsic operations on vector spaces in the ring of polynomials, this algorithm extends naturally to Laurent polynomials.

[1]  Victor Y. Pan,et al.  Controlled iterative methods for solving polynomial systems , 1998, ISSAC '98.

[2]  H. Michael Möller.  Systems of Algebraic Equations Solved by Means of Endomorphisms , 1993, AAECC.

[3]  B. Sturmfels.  Gröbner bases of toric varieties , 1991 .

[4]  J. E. Morais,et al.  When Polynomial Equation Systems Can Be "Solved" Fast? , 1995, AAECC.

[5]  H. Stetter,et al.  An Elimination Algorithm for the Computation of All Zeros of a System of Multivariate Polynomial Equations , 1988 .

[6]  Bernard Mourrain,et al.  Computing the Isolated Roots by Matrix Methods , 1998, J. Symb. Comput..

[7]  Marie-Françoise Roy,et al.  Zeros, multiplicities, and idempotents for zero-dimensional systems , 1996 .

[8]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[9]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[10]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[11]  Bernard Mourrain,et al.  Matrices in Elimination Theory , 1999, J. Symb. Comput..

[12]  Sandro Zampieri,et al.  Gröbner Bases with Respect to Generalized Term Orders and their Application to the Modelling Problem , 1996, J. Symb. Comput..

[13]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[14]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[15]  Victor Y. Pan,et al.  Multivariate Polynomials, Duality, and Structured Matrices , 2000, J. Complex..

[16]  Fabrice Rouillier,et al.  Algorithmes efficaces pour l'etude des zeros reels des systemes polynomiaux , 1996 .

[17]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

[18]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

[19]  Jean-Charles Faugère,et al.  Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering , 1993, J. Symb. Comput..

[20]  Victor Y. Pan,et al.  Asymptotic acceleration of solving multivariate polynomial systems of equations , 1998, STOC '98.

[21]  F. S. Macaulay.  On the resolution of a given modular system into primary systems including some properties of Hilbert numbers , 1913 .

[22]  Daniel Lazard,et al.  Resolution des Systemes d'Equations Algebriques , 1981, Theor. Comput. Sci..