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 be the field of complex numbers and a subvariety of . To study the ``exponential behavior of at infinity", we define as the set of limitpoints on the unit sphere of the set of real -tuples , where and . More algebraically, in the case of arbitrary base-field we can look at places ``at infinity'' on and use the values of the associated valuations on to construct an analogous set . Thirdly, simply by studying the terms occurring in elements of the ideal defining , we define another closely related set, .
These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of on , then studied further.
It is shown among other things that (when defined) . If a certain natural conjecture is true, then equality holds where we wrote ``", and the common set is a finite union of convex spherical polytopes.
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-  George M. Bergman, A weak Nullstellensatz for valuations, Proc. Amer. Math. Soc. 28 (1971), 32–38. MR 0272780, https://doi.org/10.1090/S0002-9939-1971-0272780-1
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- G. M. Bergman, A weak Nullstellensatz for valuations, Proc. Amer. Math. Soc. 28 (1971), 32-38. MR 0272780 (42:7661)
- D. Mumford, Introduction to algebraic geometry, Department of Mathematics, Harvard University, Cambridge, Mass., 1966 (preliminary version of first three chapters).
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- O. Zariski and P. Samuel, Commutative algebra. Vol. 2, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)
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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