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  "body": "\n## Summary\n\nNevanlinna theory of difference equations applies the value distribution theory of [[meromorphic functions|0ZZ7ZYDR8W]] to equations such as $f(z + 1) = R(z, f(z))$ in the complex plane.\nIts central tool is a difference analogue of the lemma on the logarithmic derivative.\nLet $f$ be meromorphic of finite order and let $c \\in \\C$.\nThen the proximity function of $f(z + c)/f(z)$ is small compared with the [[Nevanlinna characteristic|YT6Z0GWS0W]] of $f$.\nWith this, the classical methods of [[Nevanlinna theory|1EHH63VNAD]] for complex differential equations carry over to difference equations.\nIt has been proposed as a test of integrability.\nEquations with enough meromorphic solutions of finite order are expected to be discrete analogues of the Painlevé equations.\n\n## Explanations\n\n### L1 human\n\nNevanlinna theory measures how often a meromorphic function takes each value, using a growth function $T(r, f)$.\nFor differential equations, it is known that derivatives cannot make a function much bigger in this sense.\nThe difference version shows the same for shifts: $f(z + 1)$ is not much bigger than $f(z)$ on average, as long as $f$ grows at most like $e^{\\abs{z}^\\rho}$ for some $\\rho$.\nThis lets one rule out meromorphic solutions of many difference equations and single out the special, integrable ones.\n",
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