Related papers: On Schur function expansions of Thom polynomials
We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of…
This is a note on my mini-course in the International Workshop on Real and Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here we introduce a new branch of the Thom polynomial theory for singularities of…
In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape…
We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.
We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for…
Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier…
The purpose of the present paper is to investigate some structural and qualitative aspects of two different perturbations of the parameters of $g$-fractions. In this context the concept of \emph{gap} $g$-fractions is introduced. While tail…
We study certain integer valued length functions on triangulated categories and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring. The…
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed…
Special bases of orthogonal polynomials are defined, that are suited to expansions of density and potential perturbations under strict particle number conservation. Particle-hole expansions of the density response to an arbitrary…
In this paper, we study some typical arithmetic properties of Euler's totient function of polynomials over finite fields. Especially, we study polynomial analogues of some classical conjectures about Euler's totient function, such as…
We study the asymptotics of Schur polynomials with partitions $\lambda$ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots,(m-1),0)$ by at most one component at the beginning as…
We present several aspects of the "topology of meromorphic functions", which we conceive as a general theory which includes the topology of holomorphic functions, the topology of pencils on quasi-projective spaces and the topology of…
This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general…
A method is presented for using coherent vectors to calculate the explicit form of Schur polynomials which are the coefficients of Laurent expansion of a vertex operator.
In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…
The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…
We study universal polynomials of characteristic classes associated to the $\mathcal{A}$-classification (i.e. up to right-left equivalence) of holomorphic map-germs $(\mathbb{C}^2,0) \to (\mathbb{C}^n, 0)$ $(n=2,3)$. That enables us to…
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the…
In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit…