Related papers: Newton's Method as a Formal Recurrence
We show analytically that Newtonian iterations, when applied to a polynomial equation, have a positive topological entropy. In a specific example of an attempt to ``find'' the real solutions of the equation $x^2+1=0$, we show that the…
In this paper, we consider the problem of formulating the subresultant polynomials for several univariate polynomials in Newton basis. It is required that the resulting subresultant polynomials be expressed in the same Newton basis as that…
Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and $q$-shift operators can appear. Our summability criteria rely on three ingredients…
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…
In this paper we construct a generating polynomial over the rationals for the generic Newton polygon for the L function of exponential sums of the family of f = x^d+ a x^s parameterized by a, and prove some of its key properties. The…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
In this paper we prove existence of matings between a large class of renormalizable cubic polynomials with one fixed critical point and another cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our…
We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newton's Iteration. Our method exhibits quadratic convergence when…
In this article, we use Robba's method to give an estimate of the Newton polygon for the L function on torus and we can draw the Newton polygon in some special cases.
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
We compute the motivic nearby cycles of functions obtained by composition with a polynomial which is non-degenerate with respect to its Newton polyhedron. Our result involves new convolution operators and generalized nearby cycles.
Ehrhart's famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors and it turns out that the number of integral…
Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…
We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and…
In this paper, we first give a simple combinatorial proof of Tepper's identity. Then, as a by product of this interesting identity we present another proof of the well-known Wilson's identity in number theory. Finally, we obtain a…
Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…
In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…