On the Time–Space Complexity of Geometric Elimination Procedures

Abstract. In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new geometric invariant, called the degree of the input system, and the proof that the most common elimination problems have time complexity which is polynomial in this degree and the length of the input. In this paper we apply this algorithmic concept in order to exhibit an elimination procedure whose space complexity is only quadratic and its time complexity is only cubic in the degree of the input system.

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

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

[3]  D. Faddeev,et al.  Computational methods of linear algebra , 1959 .

[4]  Noaï Fitchas,et al.  Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen‐Suslin) pour le Calcul Formel , 1990 .

[5]  Marc Giusti,et al.  Lower bounds for diophantine approximations , 1997 .

[6]  Joos Heintz.  On the Computational Complexity of Polynomials and Bilinear Mappings. A Survey , 1987, AAECC.

[7]  B. Bank,et al.  Polar varieties and efficient real elimination , 2000 .

[8]  J. E. Morais,et al.  On the intrinsic complexity of the arithmetic Nullstellensatz , 2000 .

[9]  Guillermo Matera.  Sobre la complejidad en espacio y tiempo de la eliminación geométrica , 1997 .

[10]  Marc Giusti,et al.  A Gröbner Free Alternative for Polynomial System Solving , 2001, J. Complex..

[11]  Juan Rafael Sendra Pons.  Algoritmos simbólicos de Hankel en álgebra computacional , 1990 .

[12]  Klemens Haegele Hans.  Intrinsic height estimates for the nullstellensatz , 1998 .

[13]  Guillermo Matera,et al.  Probabilistic Algorithms for Geometric Elimination , 1999, Applicable Algebra in Engineering, Communication and Computing.

[14]  J. E. Morais,et al.  Straight--Line Programs in Geometric Elimination Theory , 1996, alg-geom/9609005.

[15]  Teresa Krick,et al.  A computational method for diophantine approximation , 1996 .

[16]  L. Roth.  Algebraic Surfaces , 1950, Nature.

[17]  bitnetJoos Heintz,et al.  La D Etermination Des Points Isol Es Et De La Dimension D'une Vari Et E Alg Ebrique Peut Se Faire En Temps Polynomial , 1991 .

[18]  José Enrique Morais San Miguel.  Resolución eficaz de sistemas de ecuaciones polinomiales , 1998 .

[19]  Ernst W. Mayr,et al.  Exponential space computation of Gröbner bases , 1996, ISSAC '96.

[20]  Hans-Jörg Stoß.  Lower Bounds for the Complexity of Polynomials , 1989, Theor. Comput. Sci..

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

[22]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

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

[24]  Philip Wadler,et al.  Deforestation: Transforming Programs to Eliminate Trees , 1990, Theor. Comput. Sci..

[25]  K. Ramachandra,et al.  Vermeidung von Divisionen. , 1973 .

[26]  Allan Borodin,et al.  Time Space Tradeoffs (Getting Closer to the Barrier?) , 1993, ISAAC.

[27]  Allan Borodin,et al.  The computational complexity of algebraic and numeric problems , 1975, Elsevier computer science library.

[28]  Marc Giusti,et al.  Polar Varieties, Real Equation Solving, and Data Structures: The Hypersurface Case , 1997, J. Complex..

[29]  Fabrice Rouillier,et al.  Solving Zero-Dimensional Systems Through the Rational Univariate Representation , 1999, Applicable Algebra in Engineering, Communication and Computing.

[30]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[31]  Teresa Krick,et al.  UNE APPROCHE INFORMATIQUE POUR L'APPROXIMATION DIOPHANTIENNE , 1994 .

[32]  Tetsuro Fujise,et al.  Solving Systems of Algebraic Equations by a General Elimination Method , 1988, J. Symb. Comput..

[33]  Arnold Schönhage,et al.  Fast algorithms - a multitape Turing machine implementation , 1994 .

[34]  J. M Varah,et al.  Computational methods in linear algebra , 1984 .

[35]  Allan Borodin,et al.  On Relating Time and Space to Size and Depth , 1977, SIAM J. Comput..

[36]  Ernst W. Mayr,et al.  Membership in Plynomial Ideals over Q Is Exponential Space Complete , 1989, STACS.

[37]  Kyriakos Kalorkoti.  ALGEBRAIC COMPLEXITY THEORY (Grundlehren der Mathematischen Wissenschaften 315) , 1999 .

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

[39]  J. Rafael Sendra,et al.  An Extended Polynomial GCD Algorithm Using Hankel Matrices , 1992, J. Symb. Comput..

[40]  Joos Heintz,et al.  Deformation Techniques for Efficient Polynomial Equation Solving , 2000, J. Complex..

[41]  G. A. Dirac,et al.  Moderne Algebra. I , 1951 .

[42]  Martín Sombra.  Estimaciones para el Teorema de Ceros de Hilbert , 1998 .

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

[44]  Fabrice Rouillier,et al.  Symbolic Recipes for Polynomial System Solving , 1999 .

[45]  J. E. Morais,et al.  Lower Bounds for diophantine Approximation , 1996 .

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

[47]  Joseph JáJá.  Time-Space trade-offs for some algebraic problems , 1983, JACM.

[48]  Joseph JáJá,et al.  Time-space tradeoffs for some algebraic problems , 1980, STOC '80.

[49]  Marc Giusti,et al.  The Projective Noether Maple Package: Computing the Dimension of a Projective Variety , 2000, J. Symb. Comput..

[50]  Richard Zippel,et al.  Effective polynomial computation , 1993, The Kluwer international series in engineering and computer science.

[51]  Erich Kaltofen.  Asymptotically fast solution of Toeplitz-like singular linear systems , 1994, ISSAC '94.

[52]  Joachim von zur Gathen,et al.  Parallel Arithmetic Computations: A Survey , 1986, MFCS.

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

[54]  Luis M. Pardo,et al.  How Lower and Upper Complexity Bounds Meet in Elimination Theory , 1995, AAECC.

[55]  V. Pan,et al.  Polynomial and Matrix Computations , 1994, Progress in Theoretical Computer Science.

[56]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[58]  Marc Giusti,et al.  Le rôle des structures de données dans les problèmes d'élimination , 1997 .

[59]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[60]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[61]  Jounaïdi Abdeljaoued.  Algorithmes rapides pour le calcul du polynome caracteristique , 1997 .

[62]  W. Brownawell.  Bounds for the degrees in the Nullstellensatz , 1987 .

[63]  Journal of the Association for Computing Machinery , 1961, Nature.

[64]  Alicia Dickenstein,et al.  The membership problem for unmixed polynomial ideals is solvable in single exponential time , 1991, Discret. Appl. Math..