Validating the Completeness of the Real Solution Set of a System of Polynomial Equations

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares programming to test whether a given set is "complete" with respect to the real solution set. Specifically, we test whether the Zariski closure of that set is indeed equal to the solution set of the real radical of the ideal generated by the given polynomials. Examples with finitely and infinitely many real solutions are provided, along with an example having polynomial inequalities.

[1]  A. Seidenberg.  A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA , 1954 .

[2]  M. Laurent,et al.  The approach of moments for polynomial equations , 2010 .

[3]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[4]  P. Rostalski,et al.  Semidefinite characterization and computation of real radical ideals , 2006 .

[5]  Zhonggang Zeng,et al.  Computing the multiplicity structure in solving polynomial systems , 2005, ISSAC.

[6]  B. Reznick,et al.  Sums of squares of real polynomials , 1995 .

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

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

[9]  Jonathan D. Hauenstein,et al.  Numerical Local Irreducible Decomposition , 2015, MACIS.

[10]  M. R. Osborne,et al.  Analysis of Newton’s Method at Irregular Singularities , 1983 .

[11]  Jonathan D. Hauenstein,et al.  Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry , 2013, Exp. Math..

[12]  D. Mehta,et al.  Communication: Newton homotopies for sampling stationary points of potential energy landscapes. , 2014, The Journal of chemical physics.

[13]  R. Baker Kearfott,et al.  Some tests of generalized bisection , 1987, TOMS.

[14]  Fabrice Rouillier,et al.  Finding at Least One Point in Each Connected Component of a Real Algebraic Set Defined by a Single Equation , 2000, J. Complex..

[15]  Jonathan D. Hauenstein,et al.  Numerically Solving Polynomial Systems with Bertini , 2013, Software, environments, tools.

[16]  Werner M. Seiler,et al.  Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra , 2009, Algorithms and computation in mathematics.

[17]  Monique Laurent,et al.  A unified approach to computing real and complex zeros of zero-dimensional ideals , 2009 .

[18]  Jonathan D. Hauenstein,et al.  Isosingular Sets and Deflation , 2013, Found. Comput. Math..

[19]  Wenrui Hao,et al.  Bertini_real: Software for One- and Two-Dimensional Real Algebraic Sets , 2014, ICMS.

[20]  D. Mehta,et al.  Energy-landscape analysis of the two-dimensional nearest-neighbor φ⁴ model. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Anton Leykin,et al.  Newton's method with deflation for isolated singularities of polynomial systems , 2006, Theor. Comput. Sci..

[22]  J. Hauenstein,et al.  Real solutions to systems of polynomial equations and parameter continuation , 2015 .

[23]  M. Marshall.  Positive polynomials and sums of squares , 2008 .

[24]  Pablo A. Parrilo,et al.  SOSTOOLS Version 3.00 Sum of Squares Optimization Toolbox for MATLAB , 2013, ArXiv.

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

[26]  Monique Laurent,et al.  Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals , 2008, Found. Comput. Math..

[27]  Jonathan D. Hauenstein,et al.  Regenerative cascade homotopies for solving polynomial systems , 2011, Appl. Math. Comput..

[28]  Jonathan D. Hauenstein,et al.  On Computing a Cell Decomposition of a Real Surface Containing Infinitely Many Singularities , 2014, ICMS.

[29]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[30]  Jonathan D. Hauenstein,et al.  A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations , 2009, SIAM J. Numer. Anal..

[31]  C. Kelley.  Solving Nonlinear Equations with Newton's Method , 1987 .

[32]  J. Hauenstein.  Numerically Computing Real Points on Algebraic Sets , 2011, Acta Applicandae Mathematicae.

[33]  James S. Thorp,et al.  An efficient algorithm to locate all the load flow solutions , 1993 .

[34]  Jonathan D. Hauenstein,et al.  Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety , 2014, Appl. Math. Comput..

[35]  Dan Wu,et al.  An efficient method to locate all the load flow solutions - revisited , 2015, 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[36]  Éric Schost,et al.  Polar varieties and computation of one point in each connected component of a smooth real algebraic set , 2003, ISSAC '03.

[37]  Jonathan D. Hauenstein,et al.  Numerical Computation of the Hilbert Function and Regularity of a Zero Dimensional Scheme , 2014 .

[38]  Dhagash Mehta,et al.  Exploring the potential energy landscape of the Thomson problem via Newton homotopies. , 2015, The Journal of chemical physics.

[39]  Bernard Mourrain,et al.  Certifying Isolated Singular Points and their Multiplicity Structure , 2015, ISSAC.

[40]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[41]  G. Stengle.  A nullstellensatz and a positivstellensatz in semialgebraic geometry , 1974 .

[42]  Jonathan D. Hauenstein,et al.  Certified predictor-corrector tracking for Newton homotopies , 2016, J. Symb. Comput..

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

[44]  R. Franzosi,et al.  Topological aspects of geometrical signatures of phase transitions. , 1998, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[45]  Lajos Rónyai,et al.  On the computation of matrices of traces and radicals of ideals , 2009, J. Symb. Comput..

[46]  Bican Xia,et al.  An Algorithm for Isolating the Real Solutions of Semi-algebraic Systems , 2002, J. Symb. Comput..

[47]  J. Doye,et al.  Saddle Points and Dynamics of Lennard-Jones Clusters, Solids and Supercooled Liquids , 2001, cond-mat/0108310.

[48]  Rolf Neuhaus,et al.  Computation of real radicals of polynomial ideals — II , 1998 .

[49]  Teresa Krick,et al.  An Algorithm for the Computation of the Radical of an Ideal in the Ring of Polynomials , 1991, AAECC.

[50]  Jonathan D. Hauenstein,et al.  What is numerical algebraic geometry , 2017 .

[51]  Jonathan D. Hauenstein,et al.  An a posteriori certification algorithm for Newton homotopies , 2014, ISSAC.

[52]  Jonathan D. Hauenstein,et al.  Cell decomposition of almost smooth real algebraic surfaces , 2013, Numerical Algorithms.

[53]  T. Wörmann,et al.  Radical computations of zero-dimensional ideals and real root counting , 1996 .

[54]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[55]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[56]  M. R. Osborne,et al.  Newton’s Method for Singular Problems when the Dimension of the Null Space is $>1$ , 1981 .

[57]  Fabrice Rouillier,et al.  Real Solving for Positive Dimensional Systems , 2002, J. Symb. Comput..

[58]  Charles W. Wampler,et al.  Finding All Real Points of a Complex Curve , 2006 .

[59]  Lihong Zhi,et al.  A certificate for semidefinite relaxations in computing positive-dimensional real radical ideals , 2016, J. Symb. Comput..

[60]  S. Smale.  Newton’s Method Estimates from Data at One Point , 1986 .

[61]  Bernard Mourrain,et al.  Moment matrices, border bases and real radical computation , 2011, J. Symb. Comput..

[62]  Zeng Guangxing,et al.  Computation of generalized real radicals of polynomial ideals , 1999 .

[63]  Monique Laurent,et al.  A prolongation-projection algorithm for computing the finite real variety of an ideal , 2008, Theor. Comput. Sci..

[64]  Pablo A. Parrilo,et al.  Sampling Algebraic Varieties for Sum of Squares Programs , 2015, SIAM J. Optim..

[65]  Anton Leykin,et al.  Robust Certified Numerical Homotopy Tracking , 2011, Foundations of Computational Mathematics.

[66]  Frank Sottile,et al.  ALGORITHM XXX: ALPHACERTIFIED: CERTIFYING SOLUTIONS TO POLYNOMIAL SYSTEMS , 2011 .