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Transactions of the American Mathematical Society

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



Avoidable algebraic subsets of Euclidean space

Author: James H. Schmerl
Journal: Trans. Amer. Math. Soc. 352 (2000), 2479-2489
MSC (1991): Primary 03E15, 04A20
Published electronically: July 9, 1999
MathSciNet review: 1608502
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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

Keywords: Algebraic sets, avoidable polynomials, infinite combinatorics
Received by editor(s): November 5, 1997
Published electronically: July 9, 1999
Article copyright: © Copyright 2000 American Mathematical Society