A closed solution to a special polynomial trinomial equation and semi-analytical roots for a general algebraic equation

TL;DR

This paper presents a closed-form solution for roots of a specific polynomial trinomial and offers semi-analytical roots for general algebraic equations, providing an alternative to numerical methods under certain conditions.

Contribution

It introduces a modified analytical solution for polynomial trinomials and extends Mikhalkin's integral approach to general algebraic equations, enhancing root-finding techniques.

Findings

The solution applies to polynomial equations of the form z^n + xz^{n-1} - 1=0.

Numerical examples demonstrate the effectiveness of the semi-analytical roots.

The method offers an alternative to numerical root-finding when the integral converges and is computable.

Abstract

We suggest a closed solution for the roots of polynomial trinomial algebraic equation $$z^n+xz^{n-1}-1=0$$ with an appropriate $x$. This solution is a minor modification to the work of Mikhalkin (Mikhalkin E N, 2006. On solving general algebraic equations by integrals of elementary functions, Siberian Mathematical Jounral, 47(2), 301-306). This modification, together with Mikhalkin's integral formula, provides a relatively simple analytical expression for the solution to a general algebraic equation when the polynomial coefficients are over the corresponding convergent domain. Numerical examples show that this expression can be another alternative to finding numerically the roots of a general polynomial algebraic equation when the integral involved exists and is calculated correctly.