Ideal solutions of the Tarry-Escott Problem of degree seven

TL;DR

This paper presents four new simpler parametric solutions to the degree seven Tarry-Escott problem, using quartic polynomials, improving upon the complexity of previously known solutions.

Contribution

It introduces four novel parametric ideal solutions using quartic polynomials, simplifying the existing solutions for the Tarry-Escott problem of degree seven.

Findings

Four new parametric solutions derived

Solutions are expressed by quartic polynomials

Simpler than most known solutions

Abstract

In this paper we obtain four new parametric ideal solutions of the Tarry-Escott problem of degree 7, that is, of the simultaneous diophantine equations, $\sum_{i=1}^8x_i^r=\sum_{i=1}^8y_i^r,\;r=1,\,2,\,\dots,\,7$. While all the known parametric solutions of the problem, with one exception, are given by polynomials of degrees $ \geq 5$, the solutions obtained in this paper are given by quartic polynomials, and are thus simpler than almost all of the known solutions.