Validity proof of Lazard's method for CAD construction

Abstract In 1994 Lazard proposed an improved method for cylindrical algebraic decomposition (CAD). The method comprised a simplified projection operation together with a generalized cell lifting (that is, stack construction) technique. For the proof of the method's validity Lazard introduced a new notion of valuation of a multivariate polynomial at a point. However a gap in one of the key supporting results for his proof was subsequently noticed. In the present paper we provide a complete validity proof of Lazard's method. Our proof is based on the classical parametrized version of Puiseux's theorem and basic properties of Lazard's valuation. This result is significant because Lazard's method can be applied to any finite family of polynomials, without any assumption on the system of coordinates. It therefore has wider applicability and may be more efficient than other projection and lifting schemes for CAD.

[1]  Christopher W. Brown,et al.  On delineability of varieties in CAD-based quantifier elimination with two equational constraints , 2009, ISSAC '09.

[2]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[3]  Hoon Hong,et al.  On using Lazard's projection in CAD construction , 2016, J. Symb. Comput..

[4]  G. E. Collins.  Computer Algebra of Polynomials and Rational Functions , 1973 .

[5]  Christopher W. Brown.  Improved Projection for Cylindrical Algebraic Decomposition , 2001, J. Symb. Comput..

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

[7]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[8]  Oscar Zariski,et al.  Studies in Equisingularity II. Equisingularity in Codimension 1 (and Characteristic Zero) , 1965 .

[9]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[10]  Volker Weispfenning,et al.  Simulation and Optimization by Quantifier Elimination , 1997, J. Symb. Comput..

[11]  Hoon Hong,et al.  On Lazard's Valuation and CAD Construction , 2015 .

[12]  Scott McCallum,et al.  An Improved Projection Operation for Cylindrical Algebraic Decomposition , 1985, European Conference on Computer Algebra.

[13]  Heinrich W. E. Jung.  Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x, y in der Umgebung einer Stelle x = a, y = b. , 1908 .

[14]  Hoon Hong,et al.  Testing Stability by Quantifier Elimination , 1997, J. Symb. Comput..

[15]  Shreeram S. Abhyankar,et al.  On the Ramification of algebraic functions , 1955 .

[16]  Anatolii A. Logunov,et al.  Analytic functions of several complex variables , 1965 .

[17]  Oscar Zariski,et al.  Studies in Equisingularity I Equivalent Singularities of Plane Algebroid Curves , 1965 .

[18]  Christopher W. Brown,et al.  Algorithmic methods for investigating equilibria in epidemic modeling , 2006, J. Symb. Comput..

[19]  H. Hong.  An improvement of the projection operator in cylindrical algebraic decomposition , 1990, ISSAC '90.

[20]  Scott McCallum,et al.  An Improved Projection Operation for Cylindrical Algebraic Decomposition of Three-Dimensional Space , 1988, J. Symb. Comput..

[21]  Daniel Lazard,et al.  Iterated discriminants , 2009, J. Symb. Comput..

[22]  Scott McCallum,et al.  On projection in CAD-based quantifier elimination with equational constraint , 1999, ISSAC '99.

[23]  Laurentiu Paunescu,et al.  Arcwise Analytic Stratification, Whitney Fibering Conjecture and Zariski Equisingularity , 2015, 1503.00130.

[24]  George E. Collins,et al.  Cylindrical Algebraic Decomposition I: The Basic Algorithm , 1984, SIAM J. Comput..

[25]  O Zariski.  On equimultiple subvarieties of algebroid hypersurfaces. , 1975, Proceedings of the National Academy of Sciences of the United States of America.

[26]  Daniel Lazard.  CAD and Topology of Semi-Algebraic Sets , 2010, Math. Comput. Sci..

[27]  Erich Kaltofen,et al.  Factorization of Polynomials , 1983 .

[28]  Scott McCallum.  On propagation of equational constraints in CAD-based quantifier elimination , 2001, ISSAC '01.

[29]  Bernard Mourrain,et al.  Explicit factors of some iterated resultants and discriminants , 2006, Math. Comput..

[30]  Eberhard Freitag,et al.  Analytic Functions of Several Complex Variables , 2011 .

[31]  R. J. Walker.  Algebraic curves , 1950 .

[32]  K. Hensel.  Über eine neue Theorie der algebraischen Functionen zweier Variablen , 1900 .

[33]  Adam Parusinski,et al.  The Abhyankar-Jung Theorem , 2011, 1103.2559.

[34]  G. E. Collins,et al.  Quantifier Elimination by Cylindrical Algebraic Decomposition — Twenty Years of Progress , 1998 .

[35]  Matthew England,et al.  Program Verification in the Presence of Complex Numbers, Functions with Branch Cuts etc , 2012, 2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.

[36]  Hassler Whitney,et al.  Complex Analytic Varieties , 1972 .

[37]  Christopher W. Brown,et al.  On using bi-equational constraints in CAD construction , 2005, ISSAC.

[38]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[39]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[40]  Matthew England,et al.  Truth table invariant cylindrical algebraic decomposition , 2014, J. Symb. Comput..

[41]  J. Schwartz,et al.  On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds , 1983 .