The logarithmic limitset of an algebraic variety
Author:
George M. Bergman
Journal:
Trans. Amer. Math. Soc. 157 (1971), 459469
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 basefield 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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M. Bergman, A weak Nullstellensatz for
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D. Mumford, Introduction to algebraic geometry, Department of Mathematics, Harvard University, Cambridge, Mass., 1966 (preliminary version of first three chapters).
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 Dniestr Notebook, unsolved problems in the theory of rings and modules, L. M. Andreichuk, Editor, 500 copies duplicated by the Acad. Sci. Moldavian SSR, June 1969, 6 kopeks. (Collection of 101 problems from the Kishinyev symposium on ring and module theory, 36 September, 1968. All but one of the other problems are for noncommutative or nonassociative rings. Russian.) MR 0254084 (40:7294)
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 G. M. Bergman, A weak Nullstellensatz for valuations, Proc. Amer. Math. Soc. 28 (1971), 3238. MR 0272780 (42:7661)
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 D. Mumford, Introduction to algebraic geometry, Department of Mathematics, Harvard University, Cambridge, Mass., 1966 (preliminary version of first three chapters).
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 M. Raynaud, Modèles de Néron, C. R. Acad. Sci. Paris Sér. AB 262 (1966), A345A347. MR 33 #2631.
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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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102804898
PII:
S 00029947(1971)02804898
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
