Some high degree polynomials appears in Delsarte scheme for estimating kissing numbers in $R^n$. Here is the excerpt from the [Florian Pfender and Gunter M. Ziegler. Kissing numbers, sphere packings, and some unexpected proofs][1]:

> Theorem 3 (Delsarte, Goethals and
> Seidel [11]). If $$ f(t)=\sum_{k=0}^d
> c_k G_k^{(n)}(t) $$ is a nonnegative
> combination of Gegenbauer polynomials,
> with $c_0 > 0$ and $c_k ≥ 0$
> otherwise, and if 
> $f (t) ≤ 0$ holds
> for all $t \in [−1, 1/2 ]$ , then the
> kissing number for $R^n$ is bounded by
> $$ \kappa(n)\leq \frac{f(1)}{c_0} $$

For example for $n=24$ polynomial $f_{24}(t)=(t-\frac{1}{2})(t-\frac{1}{4})^2 t^2 (t+\frac{1}{4})^2 (t+\frac{1}{2})^2(t+1)$ gives the precise number for kissing number in $R^{24}$ - 196 560.

  [1]: http://www.ams.org/notices/200408/fea-pfender.pdf