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:
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.