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.

**[1]***Dnestrovskaya tetrad: Nereshennye zadachi teorii kolets i modulei*, First All-Union Symposium on the Theory of Rings and Modules (Kishinev, 1968), Redakc.-Izdat. Otdel Akad. Nauk Moldav. SSR, Kishinev, 1969 (Russian). MR**0254084****[2]**George M. Bergman,*A weak Nullstellensatz for valuations*, Proc. Amer. Math. Soc.**28**(1971), 32–38. MR**0272780**, 10.1090/S0002-9939-1971-0272780-1**[3]**D. Mumford,*Introduction to algebraic geometry*, Department of Mathematics, Harvard University, Cambridge, Mass., 1966 (preliminary version of first three chapters).**[4]**M. Raynaud,*Modèles de Néron*, C. R. Acad. Sci. Paris Sér. A-B**262**(1966), A345-A347. MR**33**#2631.**[5]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0280489-8

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