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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polynomials on affine manifolds


Author: David Fried
Journal: Trans. Amer. Math. Soc. 274 (1982), 709-719
MSC: Primary 53C15; Secondary 57R99, 58C05
MathSciNet review: 675076
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Abstract: For a closed affine manifold $ M$ of dimension $ m$ the developing map defines an open subset $ D(\tilde M) \subset {{\mathbf{R}}^m}$. We show that $ D(\tilde M)$ cannot lie between parallel hyperplanes. When $ m \le 3$ we show that any nonconstant polynomial $ p:{{\mathbf{R}}^m} \to {\mathbf{R}}$ is unbounded on $ D(\tilde M)$. If $ D(\tilde M)$ lies in a half-space we show $ M$ has zero Euler characteristic. Under various special conditions on $ M$ we show that $ M$ has no nonconstant functions given by polynomials in affine coordinates.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0675076-0
PII: S 0002-9947(1982)0675076-0
Article copyright: © Copyright 1982 American Mathematical Society