Related papers: Existence of Bernstein-Sato polynomials by using t…
Polynomial invariants for robot manipulators and their joints arise from the adjoint action of the Euclidean group on its Lie algebra, the space of infinitesimal twists or screws. The aim of this paper is to determine basic sets of…
A family of recently investigated Bernstein functions is revisited and those functions for which the derivatives are logarithmically completely monotonic are identified. This leads to the definition of a class of Bernstein functions, which…
We give a new proof of Carlitz-Wan's conjecture, previously proved by Lenstra (1995).Our proofs are natural and intuitive, and shed new insights into the study of exceptional polynomials.
Carlitz and Scoville in 1973 considered a four variable polynomial that enumerates permutations in $\mathfrak{S}_n$ with respect to the parity of its descents and ascents. In recent work, Pan and Zeng proved a $q$-analogue of…
Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $\alpha=-m+i $- and $\beta=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually…
The conjecture on roots of Ehrhart polynomials, stated by Matsui et al. \cite[Conjecture 4.10]{MHNOH}, says that all roots $\alpha$ of the Ehrhart polynomial of a Gorenstein Fano polytope of dimension $d$ satisfy $-\frac{d}{2} \leq…
In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald's constant term identities admit an extra set of free parameters, thereby…
Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to…
In 1947 Fine obtained an expression for the number of binomial coefficients on row n of Pascal's triangle that are nonzero modulo p. In this paper we use Kummer's theorem to generalize Fine's theorem to prime powers, expressing the number…
Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable generalization with many similar properties was later considered by Dumont. They…
In a broad sense, positivstellens\"atze are results about representations of polynomials which are strictly positive on a given set. We give constructive and, to a large extent, elementary proofs of some known positivstellens\"atze for…
As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally…
It does not seem to have been observed previously that the classical Bernstein polynomials $B_N(f)(x)$ are closely related to the Bergman-Szego kernels $\Pi_N$ for the Fubini-Study metric on $\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the…
We prove the existence of signed combinatorial interpretations for several large families of structure constants. These families include standard bases of symmetric and quasisymmetric polynomials, as well as various bases in Schubert…
The main purpose of this paper is to construct not only generating functions of the new approach Genocchi type numbers and polynomials but also interpolation function of these numbers and polynomials which are related to a, b, c arbitrary…
Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and…
There are two fundamental problems motivated by Silverman's conversations over the years concerning the nature of the exact values of canonical heights of $f(z)\in\bar{\mathbb{Q}}(z)$ where $f$ has degree $d\geq 2$. The first problem is the…
We refine a result of R. Benedetto in p-adic analysis in order to exhibit infinitely many quadratic polynomials with rational coefficients having a specified graph of rational preperiodic points.
In 1853 J. Sylvester introduced a family of double sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these…
We give algorithms for computing multiplier ideals using Gr\"obner bases in Weyl algebras. The algorithms are based on a newly introduced notion which is a variant of Budur--Musta\c{t}\v{a}--Saito's (generalized) Bernstein--Sato polynomial.…