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We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
We answer the following long-standing question of Kolchin: given a system of algebraic-differential equations $\Sigma(x_1,\dots,x_n)=0$ in $m$ derivatives over a differential field of characteristic zero, is there a computable bound, that…
We generalise the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism. Instead of using weakly monotonic functionals, we interpret terms in a suitable extension of System F-omega. This…
Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
We present an approach for construction of functional bases of differential invariants for some infinite-dimensional algebras with coefficients of generating operators depending on arbitrary functions. An example for the…
We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we…
We examine multidimensional polynomial progressions involving linearly independent polynomials in finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the…
Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete…
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial…
This paper presents a unified framework for sufficient dimension reduction (SDR) that generalizes several existing SDR techniques and offers new insights into the connection between inverse conditional moment independence and dimension…
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
In this paper, we study the growth of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)\Delta^mf(z)+\cdots+P_1(z)\Delta f(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for…
In this work we present a novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems. It extends a classical subdivision technique [Dellnitz/Hohmann 1997] for the computation of such…
We construct new families of discrete vector orthogonal polynomials that have the property to be eigenfunctions of some difference operator. They are extensions of Charlier, Meixner and Kravchuk polynomial systems. The ideas behind our…
The theory part of this paper is sketched as follows. Based on column stochastic average matrix $T_n$ selected as a basic substitution matrix, the method of advanced successive difference substitution is established. Then, a set of…
The objective of this paper is, in the main, twofold: Firstly, to develop an algebraic setting for dealing with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate Stirling polynomials…