New solutions of the Tarry-Escott problem of degrees 2, 3 and 5

In this paper we obtain new parametric ideal solutions of the Tarry-Escott problem of degrees 2, 3 and 5, that is, of the diophantine systems $\sum_{i=1}^{k+1}x_i^j=\sum_{i=1}^{k+1}y_i^j,\;j=1,\,2,\,\dots,\,k$, when $k$ is 2, 3 or 5. When $k=2$, we obtain the complete ideal solution in terms of polynomials in six parameters $p, q, r, a, b$ and $c$ such that the common sums $\sigma_j=\sum_{i=1}^3x_i^j=\sum_{i=1}^3y_i^j$ for both $j=1$ and $j=2$ are symmetric functions of the parameters $p, q, r$ and also symmetric functions of the parameters $a, b, c$. When $k=3$, we obtain a solution in terms of polynomials in four parameters $p, q, r$ and $s$ such that the three common sums $\sigma_j= \sum_{i=1}^4x_i^j=\sum_{i=1}^4y_i^j, j=1, 2, 3$, are symmetric functions of all the four parameters $p, q, r$ and $s$. When $k=5$, our solution is derived from the solution already obtained when $k=2$, and the common sums, defined as in the cases when $k=2$ or 3, are either 0 or have properties similar to the case when $k=2$.