What is numerical algebraic geometry

The foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry. This article provides a short introduction to numerical algebraic geometry with the subsequent articles in this special issue considering three current research topics: solving structured systems, certifying the results of numerical computations, and performing algebraic computations numerically via Macaulay dual spaces.

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

[2]  André Galligo,et al.  Irreducible Decomposition of Curves , 2002, J. Symb. Comput..

[3]  Carlos Beltrán,et al.  On Smale's 17th Problem: A Probabilistic Positive Solution , 2008, Found. Comput. Math..

[4]  A. Morgan,et al.  Computing singular solutions to polynomial systems , 1992 .

[5]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[6]  Jan Verschelde,et al.  A Method for Tracking Singular Paths with Application to the Numerical Irreducible Decomposition , 2002 .

[7]  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..

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

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

[10]  Takeo Ojika,et al.  Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations , 1987 .

[11]  Jonathan D. Hauenstein,et al.  Certifying solutions to square systems of polynomial-exponential equations , 2011, 1109.4547.

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

[13]  Michael Shub,et al.  On the average cost of solving polynomial equations , 1983 .

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

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

[16]  H. Kuhn.  Simplicial approximation of fixed points. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Jonathan D. Hauenstein,et al.  Adaptive Multiprecision Path Tracking , 2008, SIAM J. Numer. Anal..

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

[19]  M. Hirsch.  A proof of the nonretractibility of a cell onto its boundary , 1963 .

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

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

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

[23]  E. Allgower,et al.  Numerical path following , 1997 .

[24]  Jonathan D. Hauenstein,et al.  A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems , 2015, PASCO.

[25]  T. Ojika,et al.  Deflation algorithm for the multiple roots of a system of nonlinear equations , 1983 .

[26]  Jonathan D. Hauenstein,et al.  Numerical irreducible decomposition of multiprojective varieties , 2015 .

[27]  Anton Leykin,et al.  Certified Numerical Homotopy Tracking , 2009, Exp. Math..

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

[29]  Marc Giusti,et al.  On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One , 2007, Found. Comput. Math..

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

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

[32]  Jonathan D. Hauenstein,et al.  Numerically deciding the arithmetically Cohen-Macaulayness of a projective scheme , 2016, J. Symb. Comput..

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

[34]  Lihong Zhi,et al.  Computing Isolated Singular Solutions of Polynomial Systems: Case of Breadth One , 2010, SIAM J. Numer. Anal..

[35]  Jonathan D. Hauenstein,et al.  Numerically Testing Generically Reduced Projective Schemes for the Arithmetic Gorenstein Property , 2015, MACIS.

[36]  Mauro C. Beltrametti,et al.  The Adjunction Theory of Complex Projective Varieties , 1995 .

[37]  S. Smale,et al.  Complexity of Bézout’s theorem. I. Geometric aspects , 1993 .

[38]  Mauro C. Beltrametti,et al.  A method for tracking singular paths with application to the numerical irreducible decomposition , 2002 .

[39]  Felipe Cucker,et al.  On a problem posed by Steve Smale , 2009, 0909.2114.

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

[41]  Stephen Smale,et al.  Complexity of Bezout's Theorem: III. Condition Number and Packing , 1993, J. Complex..

[42]  C. B. García,et al.  Finding all solutions to polynomial systems and other systems of equations , 1979, Math. Program..

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

[44]  Anton Leykin,et al.  Numerical Algebraic Geometry for Macaulay2 , 2009, ArXiv.

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

[46]  S. Smale,et al.  Complexity of Bezout's theorem IV: probability of success; extensions , 1996 .

[47]  Bernard Mourrain,et al.  Deflation and certified isolation of singular zeros of polynomial systems , 2011, ISSAC '11.

[48]  Andrew J. Sommese,et al.  Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems , 2002, SIAM J. Numer. Anal..

[49]  J. Yorke,et al.  A homotopy method for locating all zeros of a system of polynomials , 1979 .

[50]  Chee-Keng Yap,et al.  Mathematical software - ICMS 2014 : 4th International Congress, Seoul, South Korea, August 5-9, 2014 : proceedings , 2014 .

[51]  J. Yorke,et al.  The homotopy continuation method: numerically implementable topological procedures , 1978 .

[52]  Tien Yien Li,et al.  Mixed cell computation in Hom4PS-3 , 2017 .

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

[54]  Jonathan D. Hauenstein,et al.  Regeneration homotopies for solving systems of polynomials , 2010, Math. Comput..

[55]  Andrew J. Sommese,et al.  Homotopies for Intersecting Solution Components of Polynomial Systems , 2004, SIAM J. Numer. Anal..

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

[57]  Frank Sottile,et al.  A lifted square formulation for certifiable Schubert calculus , 2015, 1504.00979.

[58]  Anton Leykin,et al.  Higher-Order Deflation for Polynomial Systems With Isolated Singular Solutions , 2006, math/0602031.

[59]  Abraham Martín del Campo,et al.  Critical points via monodromy and local methods , 2015, 1503.01662.

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

[61]  Stephen Smale,et al.  Complexity of Bezout's Theorem V: Polynomial Time , 1994, Theor. Comput. Sci..

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

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

[64]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

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

[66]  Anton Leykin,et al.  Numerical algebraic geometry , 2020, Applications of Polynomial Systems.

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

[68]  Andrew J. Sommese,et al.  An intrinsic homotopy for intersecting algebraic varieties , 2005, J. Complex..

[69]  Frank Sottile,et al.  Galois groups of Schubert problems via homotopy computation , 2007, Math. Comput..

[70]  Herbert E. Scarf,et al.  The Solution of Systems of Piecewise Linear Equations , 1976, Math. Oper. Res..

[71]  A. Morgan,et al.  Computing singular solutions to nonlinear analytic systems , 1990 .

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

[73]  D. Mehta,et al.  Parallel degree computation for binomial systems , 2017 .

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

[75]  S. Smale,et al.  Complexity of Bezout’s Theorem II Volumes and Probabilities , 1993 .

[76]  David Rupprecht,et al.  Semi-numerical absolute factorization of polynomials with integer coefficients , 2004, J. Symb. Comput..

[77]  Tsung-Lin Lee,et al.  Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods , 2014, ICMS.

[78]  Anton Leykin,et al.  Eliminating dual spaces , 2015, 1503.02038.

[79]  Jan Verschelde,et al.  Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components , 2001 .

[80]  Andrew J. Sommese,et al.  Exceptional Sets and Fiber Products , 2008, Found. Comput. Math..

[81]  Andrew J. Newell,et al.  Decoupling Highly Structured Polynomial Systems , 2015, ACCA.

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

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

[84]  Jonathan D. Hauenstein,et al.  Witness sets of projections , 2010, Appl. Math. Comput..

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

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

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

[88]  Jonathan D. Hauenstein,et al.  Numerically intersecting algebraic varieties via witness sets , 2013, Appl. Math. Comput..

[89]  Eugene L. Allgower,et al.  Continuation and path following , 1993, Acta Numerica.

[90]  Jonathan D. Hauenstein,et al.  A Parallel Endgame , 2010 .