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Avoidable algebraic subsets of Euclidean space

Authors:	James H Schmerl

Institution:	Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Abstract:	Fix an integer $n\ge 1$ and consider real -dimensional $\mathbb{R}^n$ . A partition of $\mathbb{R}^n$ avoids the polynomial $p(x_0,x_1,\dotsc,x_{k-1})\in\mathbb Rx_0,x_1,\dotsc,x_{k-1}]$ , where each is an -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 , 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 $\mathbb R$ .

Keywords:	Algebraic sets avoidable polynomials infinite combinatorics

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