## 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