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Let $P, Q\in \mathbb{Z}[x, y]$ be polynomials with zero constant terms. Assume the induced map $\phi_{P, Q}:\mathbb{Z}^2\to\mathbb{Z}^2$ is injective. An example is $P=x+x^3 y^2, Q=y+x^2 y^3$.

Can the map $f_{P, Q}:\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$ defined by $$f_{P, Q}(n)=\sup_{|\phi_{P, Q}(v)|\leq n}|v|$$ grow faster than $2^{2^n}$?

The norm above is the $L^1$ norm.

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    $\begingroup$ Why do you think such a thing exists? $\endgroup$ Apr 17, 2021 at 9:47
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    $\begingroup$ I believe that this is a good question, but, to possibly clarify Emil's comment, you should rather ask it as "Does there exist..." (you can do so by editing the question). $\endgroup$
    – YCor
    Apr 17, 2021 at 11:56
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    $\begingroup$ Do you know about weaker versions of this question, for instance requiring that it grows faster than any polynomial? How about faster than exponential? $\endgroup$
    – Wojowu
    Apr 17, 2021 at 18:15
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    $\begingroup$ Note that the question has been edited into something very different since the earlier comments were posted. $\endgroup$ Apr 18, 2021 at 12:53
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    $\begingroup$ It might be simpler to avoid the norming, and define $f_{P,Q}(n)=\sup(\{x+y:|P(x,y)|+|Q(x,y)|\le n\})$ instead. $\endgroup$
    – user44143
    Apr 18, 2021 at 15:26

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