Polyhedral Methods in Numerical Algebraic Geometry

In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets. 2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30.

[1]  Akiko Takeda,et al.  Dynamic Enumeration of All Mixed Cells , 2007, Discret. Comput. Geom..

[2]  David H. Bailey,et al.  Integer relation detection , 2000, Computing in Science & Engineering.

[3]  A. Morgan,et al.  Coefficient-parameter polynomial continuation , 1989 .

[4]  A. Jensen.  Computing Gröbner Fans and Tropical Varieties in Gfan , 2008 .

[5]  Teo Mora,et al.  Local Parametrization of Space Curves at Singular Points , 1992 .

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

[7]  Michael Shub,et al.  Newton's method for overdetermined systems of equations , 2000, Math. Comput..

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

[9]  Jan Verschelde,et al.  Toric Newton Method for Polynomial Homotopies , 2000, J. Symb. Comput..

[10]  M. Ziegler.  Volume 152 of Graduate Texts in Mathematics , 1995 .

[11]  Shreeram S. Abhyankar,et al.  Local Analytic Geometry , 1964 .

[12]  Rekha R. Thomas.  Lectures in Geometric Combinatorics , 2006, Student mathematical library.

[13]  B. Teissier,et al.  Cl\^oture int\'egrale des id\'eaux et \'equisingularit\'e , 2008, 0803.2369.

[14]  Eecient Incremental Algorithms for the Sparse Resultant and the Mixed Volume , 1995 .

[15]  Jan Verschelde,et al.  Solving Polynomial Systems Equation by Equation , 2008 .

[16]  Jan Verschelde,et al.  Polyhedral end games for polynomial continuation , 2004, Numerical Algorithms.

[17]  J. Maurice Rojas,et al.  Toric intersection theory for affine root counting , 1996, math/9606215.

[18]  Tsung-Lin Lee,et al.  HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method , 2008, Computing.

[19]  R. Moeckel,et al.  Finiteness of relative equilibria of the four-body problem , 2006 .

[20]  Gerhard Pfister,et al.  Local analytic geometry - basic theory and applications , 2000, Advanced lectures in mathematics.

[21]  Anders Nedergaard Jensen,et al.  An algorithm for lifting points in a tropical variety , 2007, 0705.2441.

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

[23]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[24]  Tangan Gao,et al.  Algorithm 846: MixedVol: a software package for mixed-volume computation , 2005, TOMS.

[25]  J. McDonald.  Fractional Power Series Solutions for Systems of Equations , 2002, Discret. Comput. Geom..

[26]  Ariel Waissbein,et al.  Deformation Techniques for Sparse Systems , 2006, Found. Comput. Math..

[27]  V. Puiseux.  Recherches sur les fonctions algébriques. , 1850 .

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

[29]  Tien-Yien Li.  Numerical Solution of Polynomial Systems by Homotopy Continuation Methods , 2003 .

[30]  Andrew J. Sommese,et al.  Numerical Irreducible Decomposition Using PHCpack , 2003, Algebra, Geometry, and Software Systems.

[31]  B. Sturmfels,et al.  First steps in tropical geometry , 2003, math/0306366.

[32]  Jonathan D. Hauenstein,et al.  Software for numerical algebraic geometry: a paradigm and progress towards its implementation , 2008 .

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

[34]  A. Morgan,et al.  A power series method for computing singular solutions to nonlinear analytic systems , 1992 .

[35]  M. Kojima,et al.  Computing all nonsingular solutions of cyclic- n polynomial using polyhedral homotopy continuation methods , 2003 .

[36]  Rekha R. Thomas,et al.  Computing tropical varieties , 2007, J. Symb. Comput..

[37]  Dinesh Manocha,et al.  SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS , 2002 .

[38]  Adrien Poteaux,et al.  Towards a Symbolic-Numeric Method to Compute Puiseux Series: The Modular Part , 2008, ArXiv.

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

[40]  G. Ziegler.  Lectures on Polytopes , 1994 .

[41]  David H. Bailey,et al.  Numerical results on relations between fundamental constants using a new algorithm , 1989 .

[42]  Andrew J. Sommese,et al.  Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components , 2000, SIAM J. Numer. Anal..

[43]  Tien-Yien Li,et al.  Mixed Volume Computation for Semi-Mixed Systems , 2003, Discret. Comput. Geom..

[44]  J. Yorke,et al.  The cheater's homotopy: an efficient procedure for solving systems of polynomial equations , 1989 .

[45]  Akiko Takeda,et al.  PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems , 2004, Computing.

[46]  Bernard Teissier,et al.  Clôture intégrale des idéaux et équisingularité , 2008 .

[47]  Andrew J. Sommese,et al.  Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets , 2000, J. Complex..

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

[49]  J. Maurer.  Puiseux expansion for space curves , 1980 .

[50]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

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

[52]  Alicia Dickenstein,et al.  Counting solutions to binomial complete intersections , 2005, J. Complex..

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

[54]  George M. Bergman,et al.  The logarithmic limit-set of an algebraic variety , 1971 .

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

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

[57]  B. Sturmfels.  Polynomial Equations and Convex Polytopes , 1998 .

[58]  Tsung-Lin Lee,et al.  Mixed Volume Computation , A Revisit , 2007 .

[59]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[60]  J. Maurice Rojas,et al.  An optimal condition for determining the exact number of roots of a polynomial system , 1991, ISSAC '91.

[61]  B Ya Kazarnovskii.  Truncation of systems of polynomial equations, ideals and varieties , 1999 .

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

[63]  Akiko Takeda,et al.  DEMiCs: A Software Package for Computing the Mixed Volume Via Dynamic Enumeration of all Mixed Cells , 2008 .

[64]  M. Giusti,et al.  Foundations of Computational Mathematics: Kronecker's smart, little black boxes , 2001 .

[65]  Thorsten Theobald,et al.  Computing Amoebas , 2002, Exp. Math..

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

[67]  Jan Verschelde,et al.  Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation , 1999, TOMS.

[68]  J. Verschelde,et al.  Symmetrical Newton Polytopes for Solving Sparse Polynomial Systems , 1995 .

[69]  Adrien Poteaux,et al.  Good reduction of puiseux series and complexity of the Newton-Puiseux algorithm over finite fields , 2008, ISSAC '08.

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

[71]  G. Björck,et al.  Methods to divide out certain solutions from systems of algebraic equations, applied to find all cyclic 8-roots , 1994 .

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

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

[74]  R. Gardner.  Geometric Tomography: Parallel X-rays of planar convex bodies , 2006 .

[75]  Grigory Mikhalkin,et al.  Amoebas of Algebraic Varieties and Tropical Geometry , 2004, math/0403015.