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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Avoidable algebraic subsets of Euclidean space
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by James H. Schmerl PDF
Trans. Amer. Math. Soc. 352 (2000), 2479-2489 Request permission

Abstract:

Fix an integer $n\ge 1$ and consider real $n$-dimensional $\mathbb R^n$. A partition of $\mathbb R^n$ avoids the polynomial $p(x_0,x_1,\dotsc ,x_{k-1})\in \mathbb R[x_0,x_1,\dotsc ,x_{k-1}]$, where each $x_i$ is an $n$-tuple of variables, if there is no set of the partition which contains distinct $a_0,a_1,\dotsc ,a_{k-1}$ such that $p(a_0,a_1,\dotsc ,a_{k-1})=0$. The polynomial is avoidable if some countable partition avoids it. The avoidable polynomials are studied here. The polynomial $\|x-y\|^2-\|y-z\|^2$ is an especially interesting example of an avoidable one. We find (1) a countable partition which avoids every avoidable polynomial over $Q$, and (2) a characterization of the avoidable polynomials. An important feature is that both the “master” partition in (1) and the characterization in (2) depend on the cardinality of $\mathbb R$.
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Additional Information
  • James H. Schmerl
  • Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
  • MR Author ID: 156275
  • ORCID: 0000-0003-0545-8339
  • Email: schmerl@math.uconn.edu
  • Received by editor(s): November 5, 1997
  • Published electronically: July 9, 1999
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 2479-2489
  • MSC (1991): Primary 03E15, 04A20
  • DOI: https://doi.org/10.1090/S0002-9947-99-02331-4
  • MathSciNet review: 1608502