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  "body": "\n## Statement\n\nFor each positive integer $d$ there is a finite set $E(d)$ of [[prime numbers|0SN0MAEKYX]] with the following property.\nLet $p$ be a prime with $p \\notin E(d)$, and let $\\Q_p$ be the field of [[p-adic numbers|BT6T702C28]].\nLet $F$ be a [[homogeneous polynomial|2C3PAQ7H50]] of degree $d$ in $n$ variables with coefficients in $\\Q_p$.\nIf $n \u003e d^2$, then there is $x \\in \\Q_p^n$ with $x \\ne 0$ and $F(x) = 0$.\n\n## Remarks\n\nThe proof is model-theoretic.\nLet $\\varphi$ be a first-order sentence in the language of rings.\nThen $\\varphi$ holds in $\\Q_p$ for almost all primes $p$ if and only if it holds in the [[Laurent series field|YZWYF7DXTW]] $\\F_p((t))$ for almost all $p$.\nHere \"almost all\" means all but finitely many.\nThis transfer comes from [[ultraproducts|AX9Q86GB8M]] over a nonprincipal ultrafilter on the primes.\nThe ultraproducts of the fields $\\Q_p$ and of the fields $\\F_p((t))$ are [[elementarily equivalent|P3B1V182N4]] valued fields.\nBoth are Henselian with residue field of characteristic $0$, and they have the same residue field and the same value group.\nFor a fixed degree $d$, the property \"every form of degree $d$ in $d^2 + 1$ variables has a nontrivial zero\" is one first-order sentence.\nA theorem of Lang shows that it holds in every $\\F_p((t))$.\nThe finite exceptional set cannot be dropped in general: Terjanian found a form of degree $4$ in $18$ variables over $\\Q_2$ with no nontrivial zero, while $18 \u003e 4^2$.\n\n## Explanations\n\n### L1 human\n\nArtin asked whether every homogeneous polynomial of degree $d$ in more than $d^2$ variables has a nonzero root over the $p$-adic numbers.\nThe answer is no in general, but Ax and Kochen proved that it is yes for all but finitely many primes $p$, the finitely many depending on $d$.\nThe proof compares $p$-adic numbers with power series over the finite field $\\F_p$, where the answer is known, and uses logic to carry it over.\n\n### L2 human\n\nThe theorem is a celebrated application of model theory to number theory.\nThe key principle is now called the Ax–Kochen–Ershov principle.\nIt says that the first-order theory of a Henselian valued field of residue characteristic $0$ is determined by the theories of its residue field and value group.\nThe original proof gives no explicit bound for the exceptional primes.\nFor $d = 2$ and $d = 3$ the conclusion is known to hold for all primes.\n",
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