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If we have $n$ degree $d$ algebraically independent polynomials in $n$ variables in $\mathbb Z[x_1,\dots,x_n]$ then

  1. what is the maximum number of common integer solutions the system can have?

  2. same as 1. but if only $n-k$ of them are algebraically independent?

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  • $\begingroup$ I take it that by "solutions" you mean common zeroes. $\endgroup$
    – Gerry Myerson
    Commented Sep 29, 2020 at 12:46
  • $\begingroup$ 'common integer zeros'. $\endgroup$
    – Turbo
    Commented Sep 29, 2020 at 13:34
  • $\begingroup$ If there are algebraic dependencies, then there could be infinitely many common integer zeros. $\endgroup$
    – user44143
    Commented Sep 29, 2020 at 13:46
  • $\begingroup$ If the polynomials are independent, then there can be at least $d^n$ common zeroes, from the case $p_i(x_1,\ldots,x_n)=(x_i-1)\cdots(x_i-d)$. $\endgroup$
    – user44143
    Commented Sep 29, 2020 at 13:53
  • $\begingroup$ @MattF. Will $x_1x_2=m!$ work as an even more extreme case? $\endgroup$
    – Turbo
    Commented Oct 2, 2020 at 5:49

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