Elliptic Curves and Algebraic Geometry Approach in Gravity Theory II. Parametrization of a Multivariable Cubic Algebraic Equation

TL;DR

This paper extends the parametrization of multivariable cubic algebraic equations in gravity theory using complex functions related to elliptic functions, highlighting novel approaches beyond traditional algebraic geometry with potential cosmological applications.

Contribution

It introduces a new method for parametrizing multivariable cubic equations in gravity using irrational functions involving elliptic functions, expanding beyond standard two-dimensional algebraic geometry.

Findings

Parametrization of multivariable cubic equations with elliptic functions.

Application to reparametrization invariance of gravitational Lagrangian.

Potential implications for modern cosmological theories.

Abstract

In a previous paper, the general approach for treatment of algebraic equations of different order in gravity theory was exposed, based on the important distinction between covariant and contravariant metric tensor components. In the present second part of the paper it has been shown that a multivariable cubic algebraic equation can also be parametrized by means of complicated, irrational and non-elliptic functions, depending on the elliptic Weierstrass function and its derivative. As a model example, the proposed before cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been investigated. This is quite different from the standard algebraic geometry approach, where only the parametrization of two-dimensional cubic algebraic equations has been considered. Also, the possible applications in modern cosmological theories has been commented.