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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.

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The original argument is (in essence) a compactness argument, so it wouldn't construct any exceptional sets for you, or give you a bound. But perhaps other (very different) techniques give an answer? – Richard Rast Nov 28 '11 at 13:19
up vote 15 down vote accepted

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).

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Very interesting. What about lower bounds, particularly counterexamples for small $p$ and $d$? – Noam D. Elkies Nov 28 '11 at 15:42
@Noam: Apparently (haven't read the papers, they're in French) Terjanian proved that $2\in A(2k)$ for $k>1$ and $p\in A(kp(p-1))$ for $k\ge1$ and odd primes $p.$ – Charles Nov 28 '11 at 16:12
One has $p_0(4)\ge 2$ (Terjanian, 1966). In addition, the argument of Arkhipov and Karatsuba (see also Lewis and Montgomery, and Brownawell, all about 1980) shows that when $l$ is an odd prime, then $l\in A(d)$ whenever $d$ is a large enough integer divisible by $l-1$. A similar conclusion holds for $l=2$, but if I remember correctly then $d$ should be divisible by $8$. I think this argument would even show that $p_0(d)\gg d$ when $d$ is large and even (something similar is proved at the end of my paper on local solubility, 1998). As far as we know, one has $A(d)=\emptyset$ for $d$ odd. – Trevor Wooley Nov 28 '11 at 16:19
P.S. @Trevor: Welcome to Mathoverflow! – Noam D. Elkies Nov 28 '11 at 20:23

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