Related papers: Solving $p$-adic polynomial equations using Jarrat…

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…

In this paper we study the $p$-adic equation $x^q=a$ over the field of $p$-adic numbers. We construct an algorithm of calculation of criteria of solvability in the case of $q=p^m$ and present a computer program to compute the criteria for…

Newton's method for polynomial root finding is one of mathematics' most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge…

Building on a classification of zeros of cubic equations due to the $12$-th century Persian mathematician Sharaf al-Din Tusi, together with Smale's theory of {\it point estimation}, we derive an efficient recipe for computing high-precision…

This paper presents new formulary solutions for quantic polynomial equations in general forms, where we present five solutions for any fifth degree polynomial equation with real coefficients, and thereby having the possibility to calculate…

An iterative formula based on Newton Method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method…

The paper proposes a polynomial formula for solution quadratic congruences in $\mathbb{Z}_p$. This formula gives the correct answer for quadratic residue and zeroes for quadratic nonresidue. The general form of the formula for $p=3…

For an odd prime $p$, we say a polynomial $f\in \mathbb F_p[X]$ computes square roots if $f(a)^2=a$ for all nonzero, perfect squares $a\in \mathbb F_p$. When $p\equiv 3 \mod 4$, it is easy to see that $f(X)=X^{\frac{p+1}{4}}$ is the…

We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…

Given a function f: [a,b] -> R, if f(a) < 0 and f(b)> 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a,b]. Moreover, given a value-oracle for f, an approximate root of f can be computed using the…

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…

Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root…

In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More…

Several root-ratio multipoint methods for finding multiple zeros of univariate functions were recently presented. The characteristic of these methods is that they deal with $m$-th root of ratio of two functions (hence the name root-ratio…

In the article we develop Euler-Lagrange method and calculate all the roots of an arbitrary complex polynomial $P(z)$ on the base of calculation (similar to the Bernoulli-Aitken-Nikiporets methods) of the limits of ratios of Hadamard…

The purpose of this paper is to present three new methods for finding all simple zeros of polynomials simultaneously. First, we give a new method for finding simultaneously all simple zeros of polynomials constructed by applying the…

In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties…

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…

In this paper, we introduce iterated integrals associated with colored rooted trees and give proofs for the shuffle relations for $\boldsymbol{p}$-adic finite and $t$-adic symmetric polylogarithms. This method generalizes the theory of the…

We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification $X$ of $(\mathbb{C} \setminus \{0\})^n$. We homogenize the input equations to…