Related papers: Limits of quotients of real polynomial functions o…
We present an algorithm for computing limits of quotients of real analytic functions. The algorithm is based on computation of a bound on the Lojasiewicz exponent and requires the denominator to have an isolated zero at the limit point.
Necessary and sufficient conditions for the existence of limits of the form {equation*} \lim_{(x,y)\rightarrow (a,b)}\frac{f(x,y)}{g(x,y)} {equation*} are given, under the hipothesis that $f$ and $g$ are real analytic functions near the…
Given two nonzero polynomials $f, g \in\mathbb R[x,y]$ and a point $(a, b) \in \mathbb{R}^2,$ we give some necessary and sufficient conditions for the existence of the limit $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}.$…
In this work a rationalized algorithm for calculating the quotient of two complex numbers is presented which reduces the number of underlying real multiplications. The performing of a complex number division using the naive method takes 4…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…
We present criteria for deciding whether a bivariate rational function in two variables can be written as a sum of two (q-)differences of bivariate rational functions. Using these criteria, we show how certain double sums can be evaluated,…
The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…
We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.
We present six Theorems on the univariate real Polynomial, using which we develop a new algorithm for deciding the existence of atleast one real root for univariate integer Polynomials. Our algorithm outputs that no positive real root…
Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not…
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.
We describe three algorithms for computer-aided symbolic multi-loop calculations that facilitated some recent novel results. First, we discuss an algorithm to derive the canonical form of an arbitrary Feynman integral in order to facilitate…
In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.
We present a new approach for estimating the set of bifurcation values at infinity. This yields a significant shrinking of the number of coefficients in the recent algorithm introduced by Jelonek and Kurdyka for reaching critical values at…
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…
The authors' previous results on the arity gap of functions of several variables are refined by considering polynomial functions over arbitrary fields. We explicitly describe the polynomial functions with arity gap at least 3, as well as…
In 2014, Chen and Singer solved the summability problem of bivariate rational functions. Later an algorithmic proof was presented by Hou and the author. In this paper, the algorithm will be simplified and adapted to the $q$-case.
If the denominator of a rational function of several variables is sum of even powers and the numerator is a monomial, then we give a numerical criterion, using the exponents involved in the expression of the rational function, to decide if…
We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we…