A formula for the bivariate map asymptotics constants in terms of the univariate map asymptotics constants

The parameters tg, pg, tg(r) and pg(r) appear in the asymptotics for a variety of maps on surfaces and embeddable graphs. In this paper we express tg(r) in terms of tg and pg(r) in terms of pg

Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables

Analytic combinatorics studies the asymptotic behaviour of sequences through the analytic properties of their generating functions. This article provides effective algorithms required for the study of analytic combinatorics in several variables, together with their complexity analyses. Given a multivariate rational function we show how to compute its smooth isolated critical points, with respect to a polynomial map encoding asymptotic behaviour, in complexity singly exponential in the degree of its denominator. We introduce a numerical Kronecker representation for solutions of polynomial systems with rational coefficients and show that it can be used to decide several properties (0 coordinate, equal coordinates, sign conditions for real solutions, and vanishing of a polynomial) in good bit complexity. Among the critical points, those that are minimal---a property governed by inequalities on the moduli of the coordinates---typically determine the dominant asymptotics of the diagonal coefficient sequence. When the Taylor expansion at the origin has all non-negative coefficients (known as the `combinatorial case') and under regularity conditions, we utilize this Kronecker representation to determine probabilistically the minimal critical points in complexity singly exponential in the degree of the denominator, with good control over the exponent in the bit complexity estimate. Generically in the combinatorial case, this allows one to automatically and rigorously determine asymptotics for the diagonal coefficient sequence. Examples obtained with a preliminary implementation show the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201

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

Survey on counting special types of polynomials

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f is decomposable if f = g o h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f. We present a classification of all collisions at degree n = p^2 which yields an exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann (editors), Computer Algebra and Polynomials, Lecture Notes in Computer Scienc

Nonparametric Estimation and Testing of Interaction in Generalized Additive Models

The additive model overcomes the "curse of dimensionality" in general nonparametric regression problems, in the sense that it achieves the optimal rate of convergence for a one-dimensional smoother. Meanwhile, compared to the classical linear regression model, it is more flexible in defining an arbitrary smooth functional relationship between the individual regressor and the conditional mean of the response variable Y given X. However, if the true model is not additive, the estimates may be seriously biased by assuming the additive structure. In this dissertation, generalized additive models (with a known link function) are considered when containing second order interaction terms. We present an extension of the existing marginal integration estimation approach for additive models with the identity link. The corresponding asymptotic normality of the estimators is derived for the univariate component functions and interaction functions. A test statistic for testing significance of the interaction terms is developed. We obtained the asymptotics for the test functional and local power results. Monte Carlo simulations are conducted to examine the finite sample performance of the estimation and testing procedures. We code our own local polynomial pre-smoother with fixed bandwidths and apply it in the integration method. The widely used LOESS function with fixed spans is also used as a pre-smoother. Both methods provide comparable results in estimation and are shown to work well with properly chosen smoothing parameters. With a small and moderate sample size, the implementation of the test procedure based on the asymptotics may produce inaccurate results. Hence a wild bootstrap procedure is provided to get empirical critical values for the test. The test procedure performs well in fitting the correct quantiles under the null hypothesis and shows strong power against the alternative