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.
$\begingroup$
$\endgroup$
4
-
$\begingroup$ 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. $\endgroup$– Sam HopkinsJan 3, 2023 at 14:22
-
2$\begingroup$ @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 $A_i$. $\endgroup$– Jianrong LiJan 3, 2023 at 15:25
-
1$\begingroup$ 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). $\endgroup$– Sam HopkinsJan 3, 2023 at 15:39
-
$\begingroup$ @SamHopkins, thank you very much! $\endgroup$– Jianrong LiJan 3, 2023 at 15:58
Add a comment
|