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.


1. Is there an effective procedure for determining $A(d)$?
2. 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.



Determining the exceptional set in the theorem of Ax & Kochen
Scott Brown (Mem. AMS, 1978) gave a bound for the largest prime $p\_0(d)$ lying in $A(d)$. So we know that for every $d\in {\Bbb N}$, one has
$$p\_0(d)\le 2^{2^{2^{2^{2^{d^{11^{4d}}}}}}}.$$
Good! In addition, one knows that $A(d)$ is empty for $d=1$ (no prizes), $d=2$ (classical) and $d=3$ (Demyanov and Lewis, independently, about 1950). For $d=5,7,11$ and no other values, there is work of Laxton and Lewis (pre-dating Ax and Kochen) which has been made effective more recently by Leep and Yeomans, Knapp, Heath-Brown and Wooley. Thus we know that $p\_0(5)\le 13$ (Heath-Brown, 2010), and $p\_0(7)\le 883$ and $p\_0(11)\le 8053$ (Wooley, 2008).


