Deformation Techniques for Sparse Systems

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is cubic in the size of the combinatorial structure of the input system. This size is mainly represented by the cardinality and mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration.

[1]  Ariel Waissbein,et al.  Inverting bijective polynomial maps over finite fields , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Punta del Este.

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

[3]  D. N. Bernshtein.  The number of roots of a system of equations , 1975 .

[4]  A. Robinson.  I. Introduction , 1991 .

[5]  Patrice Philippon,et al.  HAUTEUR NORMALISÉE DES VARIÉTÉS TORIQUES PROJECTIVES , 2007, Journal of the Institute of Mathematics of Jussieu.

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

[7]  J. M. Rojas.  Why Polyhedra Matter in Non-Linear Equation Solving , 2002, math/0212309.

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

[9]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

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

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

[12]  Bernd Sturmfels,et al.  A polyhedral method for solving sparse polynomial systems , 1995 .

[13]  J. Canny,et al.  Efficient incremental algorithms for the sparse resultant and the mixed volume , 1995 .

[14]  John E. Savage,et al.  Models of computation - exploring the power of computing , 1998 .

[15]  Grégoire Lecerf,et al.  Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers , 2003, J. Complex..

[16]  A. G. Kushnirenko,et al.  Newton polytopes and the Bezout theorem , 1976 .

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

[18]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[19]  Teresa Krick,et al.  The Computational Complexity of the Chow Form , 2002, Found. Comput. Math..

[20]  Rudolf Lide,et al.  Finite fields , 1983 .

[21]  J. Maurice Rojas,et al.  Solving Degenerate Sparse Polynomial Systems Faster , 1998, J. Symb. Comput..

[22]  Grégoire Lecerf,et al.  A concise proof of the Kronecker polynomial system solver from scratch , 2008 .

[23]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[24]  A. Morgan.  Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems , 1987 .

[25]  岡 睦雄,et al.  Non-degenerate complete intersection singularity , 1997 .

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

[27]  J. M. Rojas.  Algebraic Geometry Over Four Rings and the Frontier to Tractability , 2000, math/0005204.

[28]  T. Y. Li.  Numerical solution of multivariate polynomial systems by homotopy continuation methods , 2008 .

[29]  José L. Balcázar,et al.  Structural Complexity I , 1995, Texts in Theoretical Computer Science An EATCS Series.

[30]  Juan Sabia,et al.  Bounds for traces in complete intersections and degrees in the Nullstellensatz , 1995, Applicable Algebra in Engineering, Communication and Computing.

[31]  Rosita Wachenchauzer,et al.  Polynomial equation solving by lifting procedures for ramified fibers , 2004, Theor. Comput. Sci..

[32]  J. Maurice Rojas,et al.  Counting Affine Roots of Polynomial Systems via Pointed Newton Polytopes , 1996, J. Complex..

[33]  Jean-Pierre Dedieu,et al.  Condition number analysis for sparse polynomial systems , 1997 .

[34]  Walter Baur,et al.  The Complexity of Partial Derivatives , 1983, Theor. Comput. Sci..

[35]  Marc Giusti,et al.  The Hardness of Polynomial Equation Solving , 2003, Found. Comput. Math..

[36]  Éric Schost,et al.  Polynomial evaluation and interpolation on special sets of points , 2005, J. Complex..

[37]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

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

[39]  Éric Schost,et al.  Tellegen's principle into practice , 2003, ISSAC '03.

[40]  Xiaoshen Wang,et al.  The BKK root count in Cn , 1996, Math. Comput..

[41]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[42]  Luis M. Pardo,et al.  Deformation techniques to solve generalised Pham systems , 2004, Theor. Comput. Sci..

[43]  Grégoire Lecerf.  Quadratic Newton Iteration for Systems with Multiplicity , 2002, Found. Comput. Math..

[44]  C. Hoffmann.  Algebraic curves , 1988 .

[45]  A. Storjohann.  Algorithms for matrix canonical forms , 2000 .

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

[47]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

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

[49]  W. Schmidt.  Equations over Finite Fields: An Elementary Approach , 1976 .

[50]  David Mumford,et al.  What Can Be Computed in Algebraic Geometry , 1993, alg-geom/9304003.

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

[52]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[53]  Marila Lázaro,et al.  De la Teoría a la Praxis: la evolución de los cursos sociales y humanísticos en una facultad de ciencias exactas y naturales , 2009 .

[54]  Bernd Sturmfels,et al.  Bernstein’s theorem in affine space , 1997, Discret. Comput. Geom..

[55]  G. Ewald.  Combinatorial Convexity and Algebraic Geometry , 1996 .

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

[57]  Timothy H. McNicholl.  Review of "Complexity and real computation" by Blum, Cucker, Shub, and Smale. Springer-Verlag. , 2001, SIGA.

[58]  Palaiseau Cedex,et al.  Computing Parametric Geometric Resolutions , 2001 .

[59]  A. Khovanskii.  Newton polyhedra and the genus of complete intersections , 1978 .

[60]  J. Verschelde,et al.  Homotopies exploiting Newton polytopes for solving sparse polynomial systems , 1994 .

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

[62]  Charles W. Wampler,et al.  A product-decomposition bound for Bezout numbers , 1995 .

[63]  Patrice Philippon,et al.  Géométrie diophantienne et variétés toriques , 2005 .

[64]  Gregorio Malajovich,et al.  High probability analysis of the condition number of sparse polynomial systems , 2004, Theor. Comput. Sci..

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

[66]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[67]  Ronald Cools,et al.  Mixed-volume computation by dynamic lifting applied to polynomial system solving , 1996, Discret. Comput. Geom..

[68]  Xing Li,et al.  Finding Mixed Cells in the Mixed Volume Computation , 2001, Found. Comput. Math..

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

[70]  Bernd Sturmfels,et al.  Product formulas for resultants and Chow forms , 1993 .

[71]  Andrew J. Sommese,et al.  The numerical solution of systems of polynomials - arising in engineering and science , 2005 .