446 research outputs found
Factoring analytic multivariate polynomials and non-standard Cauchy–Riemann conditions
Motivated by previous work on the simplification of parametrizations of curves, in this paper we generalize the well-known notion of analytic polynomial (a bivariate polynomial P (x , y ), with complex coefficients, which arises by substituting z → x + iy on a univariate polynomial View the MathML source, i.e. p (z ) → p (x + iy ) = P (x , y )) to other finite field extensions, beyond the classical case of View the MathML source. In this general setting we obtain different properties on the factorization, gcd's and resultants of analytic polynomials, which seem to be new even in the context of Complex Analysis. Moreover, we extend the well-known Cauchy–Riemann conditions (for harmonic conjugates) to this algebraic framework, proving that the new conditions also characterize the components of generalized analytic polynomials
Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
Integrable theory is formulated for correlation functions of characteristic
polynomials associated with invariant non-Gaussian ensembles of Hermitean
random matrices. By embedding the correlation functions of interest into a more
general theory of tau-functions, we (i) identify a zoo of hierarchical
relations satisfied by tau-functions in an abstract infinite-dimensional space,
and (ii) present a technology to translate these relations into hierarchically
structured nonlinear differential equations describing the correlation
functions of characteristic polynomials in the physical, spectral space.
Implications of this formalism for fermionic, bosonic, and supersymmetric
variations of zero-dimensional replica field theories are discussed at length.
A particular emphasis is placed on the phenomenon of fermionic-bosonic
factorisation of random-matrix-theory correlation functions.Comment: 62 pages, 1 table, published version (typos corrected
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
Asymptotics of Bivariate Generating Functions with Algebraic Singularities
Flajolet and Odlyzko (1990) derived asymptotic formulae the coefficients of a class of uni- variate generating functions with algebraic singularities. Gao and Richmond (1992) and Hwang (1996, 1998) extended these results to classes of multivariate generating functions, in both cases by reducing to the univariate case. Pemantle and Wilson (2013) outlined new multivariate ana- lytic techniques and used them to analyze the coefficients of rational generating functions. After overviewing these methods, we use them to find asymptotic formulae for the coefficients of a broad class of bivariate generating functions with algebraic singularities. Beginning with the Cauchy integral formula, we explicity deform the contour of integration so that it hugs a set of critical points. The asymptotic contribution to the integral comes from analyzing the integrand near these points, leading to explicit asymptotic formulae. Next, we use this formula to analyze an example from current research. In the following chapter, we apply multivariate analytic techniques to quan- tum walks. Bressler and Pemantle (2007) found a (d + 1)-dimensional rational generating function whose coefficients described the amplitude of a particle at a position in the integer lattice after n steps. Here, the minimal critical points form a curve on the (d + 1)-dimensional unit torus. We find asymptotic formulae for the amplitude of a particle in a given position, normalized by the number of steps n, as n approaches infinity. Each critical point contributes to the asymptotics for a specific normalized position. Using Groebner bases in Maple again, we compute the explicit locations of peak amplitudes. In a scaling window of size the square root of n near the peaks, each amplitude is asymptotic to an Airy function
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page
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