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Polynomials with integral coefficients, equivalent to a given polynomial
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## Polynomials with integral coefficients,

equivalent to a given polynomial

### János Kollár

**Abstract.**
Let $f(x_{0},\dots ,x_{n})$ be a homogeneous polynomial with rational
coefficients. The aim of this paper is to find a polynomial with
integral coefficients $F(x_{0},\dots ,x_{n})$ which is ``equivalent" to $f$
and as ``simple" as possible. The principal ingredient of the proof
is to connect this question with the geometric invariant theory of
polynomials. Applications to binary forms, class numbers, quadratic
forms and to families of cubic surfaces are given at the end.

*Copyright 1997 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**03** (1997), pp. 17-27
- Publisher Identifier: S 1079-6762(97)00019-X
- 1991
*Mathematics Subject Classification*. Primary 11G35, 14G25, 14D10; Secondary 11C08, 11E12, 11E76, 11R29, 14D25, 14J70
*Key words and phrases*. Polynomials, hypersurfaces, geometric invariant theory, class
numbers, quadratic forms
- Received by the editors January 30, 1997
- Posted on April 8, 1997
- Communicated by Robert Lazarsfeld
- Comments (When Available)

**János Kollár**

University of Utah, Salt Lake City, UT 84112

*E-mail address:* `kollar@math.utah.edu`

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