Cubic Algebraic Equations in Gravity Theory, Parametrization with the Weierstrass Function and Non-Arithmetic Theory of Algebraic Equations

TL;DR

This paper derives a cubic algebraic equation relevant to gravity theories, parametrizes it using the Weierstrass elliptic function with variable coefficients, and explores convergence properties of related infinite sums.

Contribution

It introduces a novel algebraic equation for gravitational Lagrangians and demonstrates its parametrization via the Weierstrass function with variable parameters, extending elliptic function theory.

Findings

The algebraic equation can be parametrized by the Weierstrass function with variable coefficients.

Certain infinite sums of inverse powers of poles are shown to converge in specific cases.

Relations are established to ensure the parametrization of the cubic equation using the Weierstrass function.

Abstract

A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity theory, brane, string and Rundall-Sundrum theories. The obtained algebraic equation was brought by means of a linear-fractional transformation to a parametrizable form, expressed through the elliptic Weierstrass function, which was proved to satisfy the standard parametrizable form, but with $g_{2}$ and $g_{3}$ functions of a complex variable instead of the definite complex numbers (known from the usual arithmetic theory of elliptic functions and curves). The generally divergent (two) infinite sums of the inverse first and second powers of the poles in the complex plane were shown to be convergent in the investigated particular case, and the case of the infinite point of the linear-fractional transformation was investigated. Some relations were found, which ensure the parametrization of the cubic equation in its general form with the Weierstrass function.