Related papers: A Note on Darboux Polynomials of Monomial Derivati…
It was recently conjectured that every system of exceptional orthogonal polynomials is related to classical orthogonal polynomials by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a…
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…
Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that…
We prove that every Diophantine quadruple in $\mathbb{R}[X]$ is regular. More precisely, we prove that if $\{a, b, c, d\}$ is a set of four non-zero polynomials from $\mathbb{R}[X]$, not all constant, such that the product of any two of its…
The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…
A Hurwitz stable polynomial of degree $n\geq1$ has a Hadamard factorization if it is a Hadamard product (i.e. element-wise multiplication) of two Hurwitz stable polynomials of degree $n$. It is known that Hurwitz stable polynomials of…
Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of…
There are no square $L^2$-flat sequences of polynomials of the type $$\frac{1}{\sqrt q}( \epsilon_0 + \epsilon_1z + \epsilon_2z^2 + \cdots + \epsilon_{q-2}z^{q-2} +\epsilon_q z^{q-1}),$$ where for each $j,~~ 0 \leq j\leq q-1,~\epsilon_j =…
In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of…
We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover,…
We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary…
We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…
One of Darboux's seminal results is archived here
It is known after Jouanolou that a general holomorphic foliation of degree $\geq2$ in projective space has no algebraic leaf. We give formulas for the degrees of the subvarieties of the parameter space of one-dimensional foliations that…
We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant…
This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results,…
In this paper we investigate to what extent the results of Z. Wang and D. Daigle on nice derivations of the polynomial ring in three variables over a field k of characteristic zero extend to the polynomial ring over a PID R, containing the…
We provide the necessary and sufficient conditions of Liouvillian integrability for Li\'{e}nard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Li\'{e}nard…
The article presents a formula expressing Macaulay constants of a numerical polynomial through its minimizing coefficients. From this, we have that Macaulay constants of Kolchin dimension polynomials do not decrease. For the minimal…
Given a non-associative unital ring $R$, a monoid $G$ and a set $\pi$ of additive maps $R \rightarrow R$, we introduce the Ore monoid ring $R[\pi ; G]$, and, in a special case, the differential monoid ring. We show that these structures…