Oppenheim conjecture
In
mathematics, the
Oppenheim conjecture is a question posed about real
quadratic forms, in the area of
diophantine approximation. It was finally settled in 1986, positively, by work of
Grigory Margulis, half a century after it had been posed initially, and after many contributions by others. The initial question was in a 1929 paper of
Alexander Oppenheim .
Let
Q be a real quadratic form subject to the following conditions:
*the number
n of variables is at least three;
*the form
Q is indefinite, in the sense that it takes both strictly positive and strictly negative values for real values of the variables;
*there is
no real number λ such that λ
Q has rational coefficients.
Then the Oppenheim conjecture states that the set of values
Q(
x1, ...,
xn)
is a
dense subset of the
real line, where the
xj are integers.
Initial work on this problem took the number
n of variables to be large, and applied a version of the
Hardy-Littlewood circle method. The definitive work of Margulis, for small values of
n, instead used methods arising from
Lie group theory.
*Oppenheim, A.
The minima of indefinite quaternary quadratic forms. Proc. Nat. Acad. Sci. U.S.A., 15:724-727, 1929.
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