Related papers: An algorithm for annihilator and Bernstein-Sato po…
The eigenvalue problem for one-dimensional Schr\"{o}dinger equation with the rational potential is numerically solved by the operator method. We show that the operator method, applied for solving the Schr\"{o}dinger equation with the…
This paper exhibits a very simple formula for a particular solution of a linear ordinary differential equation with constant real coefficients, P(d/dt)x = f, f a function given by a linear combination of polynomials, trigonometrical and…
The papers shows an algorithm to search for approximations of reals to rationals of the form a/b^2 that runs on \sqrt(b) polynomial time steps.
Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant…
We calculate the zeros of an exponential polynomial of three variables by a classical algorithm and quantum algorithms which are based on the method of van Dam and Shparlinski, they treated the case of two variables, and compare with the…
An algorithm is designed which decomposes a tropical univariate rational function into a composition of tropical binomials and trinomials. When a function is monotone, the composition consists just of binomials. Similar algorithms are…
We consider a conjectured topological inequality for the number of equisingular moduli of a rational surface singularity, and prove it in some natural special cases. When the resolution dual graph is "sufficiently negative" (in a precise…
In this work we present a method, based on the use of Bernstein polynomials, for the numerical resolution of some boundary values problems. The computations have not need of particular approximations of derivatives, such as finite…
The extended L\"uroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In…
Artin solved Hilbert's $17^{th}$ problem by showing that every positive semidefinite polynomial can be realized as a sum of squares of rational functions. Pfister gave a bound on the number of squares of rational functions: if $p$ is a…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…
Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincar\'e series (zeta function).…
We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near…
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a…
A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.
Given a properly normalized parametrization of a genus-0 modular curve, the complex multiplication points map to algebraic numbers called singular moduli. In the classical case, the maps can be given analytically. However, in the Shimura…
In quantum theory, the so-called "spinless Salpeter equation," the relativistic generalization of the nonrelativistic Schroedinger equation, is used to describe both bound states of scalar particles and the spin-averaged spectra of bound…
In this paper, we construct a family of Bernstein functions using a class of rational parametrization. The new family of rational Bernstein basis on an index $\alpha \in {\left(-\infty \, , \, 0 \right)}\cup {\left(1 \, , \,…
We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the…
A new type of combinations of Bernstein operators is given in [1]. Here, we introduce another one, which can be used to approximate the functions with singularities. The direct and inverse results of the weighted approximation of this new…