Related papers: Joint differential resolvents for pseudopolynomial…

Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…

An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this…

We construct biorthogonal polynomials for a measure over the complex plane which consists in the exponential of a potential V(z,z*) and in a set of external sources at the numerator and at the denominator. We use the pseudonorm of these…

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting…

In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: $f^{n}+P(f)=R(z)e^{\alpha(z)}$ and $f^{n}+P_{*}(f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$ in the complex plane, where…

Building on the work of Arizmendi and Celestino (2021), we derive the $*$-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers $\alpha$ and $\beta$, we derive…

We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case…

Differential resultant formulas are defined, for a system $\mathcal{P}$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials…

We introduce a non-commutative resultant, for slice regular polynomials in two quaternionic variables, defined in terms of a suitable Dieudonn\'e determinant.We use this tool to investigate the existence of common zeros of slice regular…

In this paper, we fix a polynomial with complex coefficients and determine the eigenforms for SL2(Z) which can be expressed as the fixed polynomial evaluated at other eigenforms. In particular, we show that when one excludes trivial cases,…

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establishe an analogue of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a…

Let $\cP$ be a system of $n$ linear nonhomogeneous ordinary differential polynomials in a set $U$ of $n-1$ differential indeterminates. Differential resultant formulas are presented to eliminate the differential indeterminates in $U$ from…

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…

Sharp resolvent bounds for non-selfadjoint semiclassical elliptic quadratic differential operators are established, in the interior of the range of the associated quadratic symbol.

The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear…

We propose a level proximal subdifferential for a proper lower semicontinuous function. Level proximal subdifferential is a uniform refinement of the well-known proximal subdifferential, and has the pleasant feature that its resolvent…

The classical multidimensional resultant can be defined as the, suitably normalized, generator of a projective elimination ideal in the ring of universal coefficients. This is the approach via the so-called inertia forms or…

Let \( P(z) \) be a polynomial of degree \( n \) and $\alpha \in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_\alpha P(z) \) and is defined by $D_\alpha P(z): = nP(z) + (\alpha -z)P'(z)$. The polar derivative \( D_\alpha…

We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.

Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…