Extremal Real Algebraic Geometry and A-Discriminants

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the A-discriminant, and give new bounds on the topology of certain A-discriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations, e.g., the number of smooth topological types attainable in certain families of real algebraic surfaces.

[1]  Michel Coste.  Topological types of fewnomials , 1998 .

[2]  Richard E. Ewing,et al.  "The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics" , 1986 .

[3]  Luis M. Pardo,et al.  Systems of rational polynomial equations have polynomial size approximate zeros on the average , 2003, J. Complex..

[4]  S. Basu,et al.  On the number of homotopy types of fibres of a definable map , 2006, math/0605517.

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

[6]  Nicolai Vorobjov,et al.  Betti Numbers of Semialgebraic and Sub‐Pfaffian Sets , 2004 .

[7]  Daniel Perrucci.  Some Bounds for the Number of Components of Real Zero Sets of Sparse Polynomials , 2005, Discret. Comput. Geom..

[8]  Frederic Bihan amd Frank Sottile.  NEW FEWNOMIAL UPPER BOUNDS FROM GALE DUAL POLYNOMIAL SYSTEMS , 2006 .

[9]  V. Kaloshin.  The Existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles , 2003 .

[10]  Pascal Koiran.  Randomized and deterministic algorithms for the dimension of algebraic varieties , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[11]  M. Kapranov,et al.  A characterization ofA-discriminantal hypersurfaces in terms of the logarithmic Gauss map , 1991 .

[12]  V. Kharlamov,et al.  Asymptotic Growth of the Number of Classes of Real Plane Algebraic Curves as the Degree Grows , 2003 .

[13]  Thomas Lickteig,et al.  Sylvester-Habicht Sequences and Fast Cauchy Index Computation , 2001, J. Symb. Comput..

[14]  Paula B. Cohen,et al.  Fewnomials and Intersections of Lines with Real Analytic Subgroups in Gmn , 2002 .

[15]  A. Wilkie.  TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248) By L OU VAN DEN D RIES : 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), ISBN 0 521 59838 9 (Cambridge University Press, 1998). , 2000 .

[16]  Bertrand Haas.  A Simple Counterexample to Kouchnirenko ’ s Conjecture , 2002 .

[18]  Ron Goldman,et al.  Topics in Algebraic Geometry and Geometric Modeling , 2003 .

[19]  Dima Grigoriev,et al.  Complexity of Quantifier Elimination in the Theory of Algebraically Closed Fields , 1984, MFCS.

[20]  J. Maurice Rojas,et al.  First Steps in Algorithmic Fewnomial Theory , 2004, math/0411107.

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

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

[23]  Alicia Dickenstein,et al.  Some results on inhomogeneous discriminants , 2006 .

[24]  J. Brasselet.  Introduction to toric varieties , 2004 .

[25]  Alicia Dickenstein,et al.  Tropical Discriminants , 2005, math/0510126.

[26]  L. van den Dries,et al.  Tame Topology and O-minimal Structures , 1998 .

[27]  J. Maurice Rojas,et al.  New Complexity Bounds for Certain Real Fewnomial Zero Sets , 2007, ArXiv.

[28]  J. Maurice Rojas,et al.  On solving univariate sparse polynomials in logarithmic time , 2005, J. Complex..

[29]  A. Storjohann.  Computing Hermite and Smith normal forms of triangular integer matrices , 1998 .

[30]  F. Loeser.  Polytopes secondaires et discriminants , 1991 .

[31]  Dmitry Grigoriev,et al.  Complexity of gene circuits, Pfaffian functions and the morphogenesis problem , 2003 .

[32]  S. Smale.  Mathematical problems for the next century , 1998 .

[33]  Nicolai Vorobjov,et al.  Complexity of cylindrical decompositions of sub-Pfaffian sets , 2001 .

[34]  Marie-Françoise Roy.  Basic algorithms in real algebraic geometry and their complexity: from Sturm's theorem to the existential theory of reals , 1996 .

[35]  A. Wilkie.  A theorem of the complement and some new o-minimal structures , 1999 .

[36]  W. Habicht.  Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens , 1948 .

[37]  J. Maurice Rojas.  Arithmetic multivariate Descartes' rule , 2001 .

[38]  A. Tsikh,et al.  Algebraic Equations and Hypergeometric Series , 2004 .

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

[40]  Joachim Rosenthal,et al.  Some remarks on real and complex output feedback , 1997 .

[41]  David A. Plaisted,et al.  New NP-hard and NP-complete polynomial and integer divisibility problems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[42]  Ragni Piene,et al.  The Legacy of Niels Henrik Abel , 2004 .

[43]  A. Harnack,et al.  Ueber die Vieltheiligkeit der ebenen algebraischen Curven , 1876 .

[44]  J. Maurice Rojas,et al.  Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces , 2003, Discret. Comput. Geom..

[45]  Mikael Passare,et al.  Amoebas: their spines and their contours , 2005 .

[46]  Alicia Dickenstein,et al.  Elimination Theory in Codimension 2 , 2002, J. Symb. Comput..