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We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the…
Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and…
We consider a nonhomogeneous Burgers equation with time variable coefficients, and obtain an explicit solution of the general initial value problem in terms of solution to a corresponding linear ODE. Special exact solutions such as…
General factors are a generalization of matchings. Given a graph $G$ with a set $\pi(v)$ of feasible degrees, called a degree constraint, for each vertex $v$ of $G$, the general factor problem is to find a (spanning) subgraph $F$ of $G$…
We study monogeneity in {\sl period equations}, $\psi_e(x)$, the auxiliary equations introduced by Gauss to solve cyclotomic polynomials by radicals. All monogenic $\psi_e(x)$ of degrees $4 \leq e \leq 250$ are determined for extended…
This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed,…
In this article, we study a boundary value problem of a class of singular linear discrete time systems whose coefficients are non-square constant matrices or square with a matrix pencil which has an identically zero determinant. By taking…
To cater to the needs of (Zero Knowledge) proofs for (mathematical) proofs, we describe a method to transform formal sentences in 2x2-matrices over multivariate polynomials with integer coefficients, such that usual proof-steps like…
We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of Boyd--Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure…
The main theme of the article is the study of discrete systems of material points subjected to constraints not only of a geometric type (holonomic constraints) but also of a kinematic type (nonholonomic constraints). The setting up of the…
It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…
Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$…
We study the algebraic dynamical systems generated by triangular systems of rational functions and estimate the height growth of iterations generated by such systems. Further, using a result on the reduction modulo primes of systems of…
This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming,…
One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both…
We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.
The derivative of a polynomial with all zeros on the unit circle has the zeros of its derivative on or inside the unit circle. It has been observed that in many cases the zeros of the derivative have a bimodal distribution: there are two…
We establish a new class of integrable {\it systems of Kowalevski type}, associated with discriminantly separable polynomials of degree two in each of three variables. Defining property of such polynomials, that all discriminants as…
Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach:…
Following the author's previous works, we continue to consider the problem of counting the number of affine conjugacy classes of polynomials of one complex variable when its unordered collection of holomorphic fixed point indices is given.…