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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The logarithmic limit-set of an algebraic variety

Author: George M. Bergman
Journal: Trans. Amer. Math. Soc. 157 (1971), 459-469
MSC: Primary 14.01
MathSciNet review: 0280489
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Abstract: 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 \vert{x_1}\vert, \ldots ,{u_x}\log \vert{x_n}\vert)$, where $ x \in V$ and $ {u_x} = {(1 + \Sigma {(\log \vert{x_i}\vert)^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.

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Keywords: Variety, exponential behavior at infinity, valuation, place, convex polytope, general linear group, dimension, logarithm, absolute value, sphere at infinity
Article copyright: © Copyright 1971 American Mathematical Society