On mixed polynomials of bidegree (n, 1)

Specifying the bidegrees (n,m) of mixed polynomials P(z,z¯) of the single complex variable z, with complex coefficients, allows to investigate interesting roots structures and counting; intermediate between complex and real algebra. Multivariate mixed polynomials appeared in recent papers dealing with Milnor fibrations, but in this paper we focus on the univariate case and m=1, which is closely related to the important subject of harmonic maps. Here we adapt, to this setting, two algorithms of computer algebra: Vandermonde interpolation and a bissection-exclusion method for root isolation. Implemented in Maple, they are used to explore some interesting classes of examples.

[1]  Bernard Mourrain,et al.  Generalized Resultants over Unirational Algebraic Varieties , 2000, J. Symb. Comput..

[2]  L. Fialkow,et al.  Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation , 2003 .

[3]  H. Weyl.  Randbemerkungen zu Hauptproblemen der Mathematik , 1924 .

[4]  Satya N. Majumdar,et al.  Condensation of the Roots of Real Random Polynomials on the Real Axis , 2009 .

[5]  G. C. Sih,et al.  Complex variable methods in elasticity , 1971 .

[6]  J. Milnor.  Singular points of complex hypersurfaces , 1968 .

[7]  Lukas Geyer.  Sharp bounds for the valence of certain harmonic polynomials , 2005 .

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

[9]  Philip J. Davis.  The Schwarz Function and Its Applications: Properties in the Large of the Schwarz Function , 1974 .

[10]  Walter Bergweiler,et al.  On the Number of Solutions of a Transcendental Equation Arising in the Theory of Gravitational Lensing , 2009, 0908.4595.

[11]  José Seade,et al.  On Real Singularities with a Milnor Fibration , 2002 .

[12]  Leonie Moench,et al.  Some Basic Problems Of The Mathematical Theory Of Elasticity , 2016 .

[13]  R. Araujo dos Santos.  Equivalence of real Milnor fibrations for quasi-homogeneous singularities , 2012 .

[14]  Youkow Homma,et al.  Counting Zeros of Harmonic Rational Functions and Its Application to Gravitational Lensing , 2012, 1206.2273.

[15]  Dmitry Khavinson,et al.  Transcendental Harmonic Mappings and Gravitational Lensing by Isothermal Galaxies , 2009, 0908.3310.

[16]  Brian Rider,et al.  A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula , 2008, 0807.4984.

[17]  T. SHEIL-SMALL,et al.  Complex Polynomials: Contents , 2002 .

[18]  B. Laurent,et al.  Implicit matrix representations of rational Bézier curves and surfaces , 2014 .

[19]  H. Shapiro.  The Schwarz Function and Its Generalization to Higher Dimensions , 1992 .

[20]  Mutsuo Oka,et al.  Non-degenerate mixed functions , 2009, 0909.1904.

[21]  Laurent Busé,et al.  Implicit matrix representations of rational Bézier curves and surfaces , 2014, Comput. Aided Des..

[22]  P. Davis.  The Schwarz function and its applications , 1974 .

[23]  André Galligo.  Deformation of roots of polynomials via fractional derivatives , 2013, J. Symb. Comput..

[24]  Jean-Claude Yakoubsohn,et al.  Numerical analysis of a bisection-exclusion method to find zeros of univariate analytic functions , 2005, J. Complex..

[25]  A. Petters,et al.  Mathematics of gravitational lensing: multiple imaging and magnification , 2009, 0912.0490.

[26]  A. S. Wilmshurst.  The valence of harmonic polynomials , 1998 .

[27]  Jürgen Moser,et al.  Normal forms for real surfaces in C2 near complex tangents and hyperbolic surface transformations , 1983 .

[28]  Dmitry Khavinson,et al.  On the number of zeros of certain harmonic polynomials , 2002 .

[29]  Dmitry Khavinson,et al.  On the number of zeros of certain rational harmonic functions , 2004, math/0401188.

[30]  Sun Hong Rhie.  n-point Gravitational Lenses with 5(n-1) Images , 2003 .

[31]  Mutsuo Oka,et al.  INTERSECTION THEORY ON MIXED CURVES , 2011, 1104.3386.

[32]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[33]  Dmitry Khavinson,et al.  Gravitational lensing by a collection of objects with radial densities , 2011 .

[34]  Seung-Yeop Lee,et al.  Remarks on Wilmshurst's theorem , 2013 .

[35]  André Galligo,et al.  Exploring univariate mixed polynomials , 2014, SNC.

[36]  C. Fassnacht,et al.  Gravitational Lensing by Elliptical Galaxies, and the Schwarz Function , 2007, 0708.2684.