Nullstellensatz for amoebas

Let $f(x_1,\ldots,x_n)$ be a Laurent polynomial, and write $f(x_1,\ldots,x_n) = \sum_{n=1}^l m_i(x)$, where $m_i(x)$ are the monomial terms of $x$. Given a point $a \in \mathbb{R}^n$, let $f\{a\}$ denote the list of positive reals $[ \vert m_1(\mathrm{Log}^{-1}(a))\vert , \ldots , \vert m_l(\mathrm{Log}^{-1}(a))\vert ]$. Note this is well defined, even though $\mathrm{Log}$ is not injective.

We say that a list of positive numbers satisfies the polygon condition if it is possible to make a polygon with those side lengths, i.e. no number is greater than the sum of all the others.

Theorem 1. Let $I$ be an ideal, and $A(I)$ its amoeba. Then $a \in A(I)$ if and only if $f\{a\}$ satisfies the polygon condition for all $f \in I$.

Let $P(f) = \{a \in \mathbb{R}^n \, : \, f\{a\} \text{ satisfies\ the\ polygon\ condition}\}$. Think of this as an approximation to the amoeba of a hypersurface.

Theorem 2. Let $A(f)$ be the amoeba of a hypersurface. Let

\begin{displaymath} f_m(x_1,\ldots,x_n) = \text{the\ product\ of } f(u_1 x_1, \ldots, u_n x_n) \end{displaymath}

over all $u_i$ such that $u_i^m = 1$. The family $P(f_m)$ converges uniformly (in the Euclidean norm) to $A(f)$.


Questions:

  1. Is there a version of theorem 2 (an explicit family approximating the amoeba) in the higher codimension case?

  2. An analogous statement to theorem 1 is known for non-archimedean Amoebas. Is theorem 1 true in an even more general context?

  3. The convergence of the family in theorem 2 is of order $O(\log m/m)$, at least in worst case situations. How fast does this family converge for a randomly chosen $f$? If the approximation is within $(a \log m + b)/m$ of the actual amoeba, what are $a$ and $b$, in typical examples?

  4. What open problems can this be used to solve?

(contributed by Kevin Purbhoo)



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