On the complexity of computing a Gröbner basis for the radical of a zero dimensional ideal

We show that if a system of polynomials f l , f 2 , . . . ,Jr in n variables with deg(fl) _< d over the rational numbers has only finitely many affine zeros, then, all the affine zeros can be determined in time polynomial in d n by a Las Vegas type randomized algorithm. We then describe single exponential time algorithms to compute reduced Gr5bner bases for the radical of the ideal generated by fi and for all the prime ideals containing the radical.

[1]  John Canny,et al.  The complexity of robot motion planning , 1988 .

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

[3]  B. Buchberger,et al.  Grobner Bases : An Algorithmic Method in Polynomial Ideal Theory , 1985 .

[4]  Erich Kaltofen,et al.  Solving systems of nonlinear polynomial equations faster , 1989, ISSAC '89.

[5]  Joseph F. Traub,et al.  On Euclid's Algorithm and the Theory of Subresultants , 1971, JACM.

[6]  James Renegar.  On the Worst-Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials , 1989, SIAM J. Comput..

[7]  Patrizia M. Gianni,et al.  Gröbner Bases and Primary Decomposition of Polynomial Ideals , 1988, J. Symb. Comput..

[8]  Patrizia M. Gianni,et al.  Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases , 1987, AAECC.

[9]  George E. Collins,et al.  Subresultants and Reduced Polynomial Remainder Sequences , 1967, JACM.

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

[11]  Franz Winkler,et al.  A p-Adic Approach to the Computation of Gröbner Bases , 1988, J. Symb. Comput..

[12]  Marc Giusti,et al.  Some Effectivity Problems in Polynomial Ideal Theory , 1984, EUROSAM.

[13]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

[14]  Dexter Kozen,et al.  Parallel Resultant Computation , 1990 .

[15]  David Shannon,et al.  Using Gröbner Bases to Determine Algebra Membership Split Surjective Algebra Homomorphisms Determine Birational Equivalence , 1988, J. Symb. Comput..

[16]  George E. Collins,et al.  The Calculation of Multivariate Polynomial Resultants , 1971, JACM.

[17]  John F. Canny,et al.  Generalized Characteristic Polynomials , 1988, ISSAC.

[18]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[19]  Wolfgang Trinks,et al.  Über B. Buchbergers verfahren, systeme algebraischer gleichungen zu lösen , 1978 .

[20]  A. Seidenberg.  Constructions in algebra , 1974 .

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

[22]  Erich Kaltofen,et al.  Improved Sparse Multivariate Polynomial Interpolation Algorithms , 1988, ISSAC.

[23]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[24]  Y. N. Lakshman,et al.  On the Complexity of Zero-dimensional Algebraic Systems , 1991 .

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

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

[27]  Michael Eugene Stillman,et al.  On the Complexity of Computing Syzygies , 1988, J. Symb. Comput..

[28]  M. Reid,et al.  Commutative ring theory: Regular rings , 1987 .

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