Polynomials with integral coefficients, equivalent to a given polynomial

Kollár, János

doi:10.1090/S1079-6762-97-00019-X

Polynomials with integral coefficients, equivalent to a given polynomial

Author: János Kollár
Journal: Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 17-27
MSC (1991): Primary 11G35, 14G25, 14D10; Secondary 11C08, 11E12, 11E76, 11R29, 14D25, 14J70
DOI: https://doi.org/10.1090/S1079-6762-97-00019-X
Published electronically: April 8, 1997
MathSciNet review: 1445631
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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.

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