Ideal solutions of the Tarry-Escott Problem of degree seven

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.