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Let $A_1 \subset A_2 \subset \cdots$ and each $A_i$ is a finite set of polynomials over variables $x_1, \ldots, x_n$. For each $i$, let $N_i$ be the Newton polytope of $A_i$. Since $A_{\infty}$ has infinite many polynomials, maybe we cannot define $N_{\infty}$ to be the Newton polytope of $A_{\infty}$ directly. Is there some way to define $N_{\infty}$ rigorously? Thank you very much.

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  • $\begingroup$ How do you define the Newton polytope of a set of polynomials? I am used to the Newton polytope being defined by a single polynomial. $\endgroup$ Jan 3, 2023 at 14:22
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    $\begingroup$ @SamHopkins, thank you very much. I take $N_i$ to be the Newton polytope of the polynomial obtained as the product of the polynomials in $A_i$. $\endgroup$ Jan 3, 2023 at 15:25
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    $\begingroup$ Ah I see. This is the same as Minkowski sum of the individual Newton polytopes. So if you want to take a limit of Minkowski sums, you could try for a "Minkowski integral" in the sense of Billera--Sturmfels (doi.org/10.2307/2946575). $\endgroup$ Jan 3, 2023 at 15:39
  • $\begingroup$ @SamHopkins, thank you very much! $\endgroup$ Jan 3, 2023 at 15:58

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