Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}_p$ in at least $d^2+1$ variables has a nontrivial zero.

Is there an effective procedure for determining $A(d)$?

For what values of $d$ is $A(d)$ known? Ax & Kochen mention that the special cases $d\in\{2,3,5,7,11\}$ were known but not if $A$ was known for those cases.

[1] James Ax and Simon Kochen, "Diophantine problems over local fields I.", *American Journal of Mathematics* **87** (1965), pp. 605–630.

[2] Simon Kochen, "The model theory of local fields", *Lecture Notes in Mathematics* **499** (1975), pp. 384–425.