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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Polynomials on affine manifolds
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by David Fried
Trans. Amer. Math. Soc. 274 (1982), 709-719
DOI: https://doi.org/10.1090/S0002-9947-1982-0675076-0

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.
References
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Bibliographic Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 274 (1982), 709-719
  • MSC: Primary 53C15; Secondary 57R99, 58C05
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0675076-0
  • MathSciNet review: 675076