The logarithmic limit-set of an algebraic variety

Let $C$ be the field of complex numbers and $V$ a subvariety of ${(C - \{ 0\} )^n}$. To study the “exponential behavior of $V$ at infinity", we define $V_\infty ^{(a)}$ as the set of limitpoints on the unit sphere ${S^{n - 1}}$ of the set of real $n$-tuples $({u_x}\log |{x_1}|, \ldots ,{u_x}\log |{x_n}|)$, where $x \in V$ and ${u_x} = {(1 + \Sigma {(\log |{x_i}|)^2})^{ - 1/2}}$. More algebraically, in the case of arbitrary base-field $k$ we can look at places “at infinity” on $V$ and use the values of the associated valuations on ${X_1}, \ldots ,{X_n}$ to construct an analogous set $V_\infty ^{(b)}$. Thirdly, simply by studying the terms occurring in elements of the ideal $I$ defining $V$, we define another closely related set, $V_\infty ^{(c)}$. These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of $GL(n,Z)$ on $k[X_1^{ \pm 1}, \ldots ,X_n^{ \pm 1}]$, then studied further. It is shown among other things that $V_\infty ^{(b)} = V_\infty ^{(c)} \supseteq$ (when defined) $V_\infty ^{(a)}$. If a certain natural conjecture is true, then equality holds where we wrote “$\supseteq$", and the common set ${V_\infty } \subseteq {S^{n - 1}}$ is a finite union of convex spherical polytopes.