Exponential space computation of Gröbner bases

Given a polynomial ideal and a term order, there is a unique reduced Groebner basis and, for each polynomial, a unique normal form, namely the smallest (w.r.t. the term order) polynomial in the same coset. We consider the problem of finding this normal form for any given polynomial, without prior computation of the Groebner basis. This is done by transforming a representation of the normal form into a system of linear equations and solving this system. Using the ability to find normal forms, we show how to obtain the Groebner basis in exponential space.

[1]  Ernst W. Mayr,et al.  An optimal algorithm for constructing the reduced Gröbner basis of binomial ideals , 1996, ISSAC '96.

[2]  L. Csanky,et al.  Fast parallel matrix inversion algorithms , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[3]  L. Csanky,et al.  Fast Parallel Matrix Inversion Algorithms , 1976, SIAM J. Comput..

[4]  K. Mulmuley.  A fast parallel algorithm to compute the rank of a matrix over an arbitrary field , 1987, Comb..

[5]  Oscar H. Ibarra,et al.  A Note on the Parallel Complexity of Computing the Rank of Order n Matrices , 1980, Inf. Process. Lett..

[6]  Bruno Buchberger,et al.  A criterion for detecting unnecessary reductions in the construction of Groebner bases , 1979, EUROSAM.

[7]  André Galligo,et al.  Some New Effectivity Bounds in Computational Geometry , 1988, AAECC.

[8]  Dung T. Huynh,et al.  A Superexponential Lower Bound for Gröbner Bases and Church-Rosser Commutative Thue Systems , 1986, Inf. Control..

[9]  J. Hopcroft,et al.  Fast parallel matrix and GCD computations , 1982, FOCS 1982.

[10]  Victor Y. Pan,et al.  Complexity of Parallel Matrix Computations , 1987, Theor. Comput. Sci..

[11]  Thomas Dubé,et al.  The Structure of Polynomial Ideals and Gröbner Bases , 2013, SIAM J. Comput..

[12]  Franco P. Preparata,et al.  An Improved Parallel Processor Bound in Fast Matrix Inversion , 1978, Inf. Process. Lett..

[13]  Stuart J. Berkowitz,et al.  On Computing the Determinant in Small Parallel Time Using a Small Number of Processors , 1984, Inf. Process. Lett..

[14]  Steven Fortune,et al.  Parallelism in random access machines , 1978, STOC.

[15]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[16]  Ketan Mulmuley,et al.  A fast parallel algorithm to compute the rank of a matrix over an arbitrary field , 1986, STOC '86.

[17]  Volker Weispfennig,et al.  Constructing universal Groebner bases , 1987 .

[18]  Victor Y. Pan,et al.  Parallel Evaluation of the Determinant and of the Inverse of a Matrix , 1989, Inf. Process. Lett..

[19]  H. Michael Möller,et al.  Upper and Lower Bounds for the Degree of Groebner Bases , 1984, EUROSAM.

[20]  Grete Hermann,et al.  Die Frage der endlich vielen Schritte in der Theorie der Polynomideale , 1926 .

[21]  Teresa Krick,et al.  Membership problem, Representation problem and the Computation of the Radical for one-dimensional Ideals , 1991 .

[22]  Allan Borodin,et al.  Fast parallel matrix and GCD computations , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[23]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[24]  Ernst W. Mayr,et al.  On Polynomial Ideals, Their Complexity, and Applications , 1995, FCT.

[25]  Joos Heintz,et al.  On the Intrinsic Complexity of Elimination Theory , 1993, J. Complex..

[26]  Lyn J. Miller.  Analogs of Gröbner Bases in Polynomial Rings over a Ring , 1996, J. Symb. Comput..

[27]  Teo Mora,et al.  The Gröbner Fan of an Ideal , 1988, J. Symb. Comput..

[28]  Lorenzo Robbiano,et al.  On the Theory of Graded Structures , 1986, J. Symb. Comput..

[29]  Bruno Buchberger,et al.  A note on the complexity of constructing Gröbner-Bases , 1983, EUROCAL.

[30]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[31]  F. S. Macaulay,et al.  The Algebraic Theory of Modular Systems , 1972 .