Abstract. Given any positive integer M , we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms with even cross coefficients in M variables.
Abstract. For any positive integer M we show that there are infinitely many real quadratic fields that do not admit M -ary universal quadratic forms (without any restriction on the parity of their cross coefficients).
We study totally positive, additively indecomposable integers in a real
quadratic field $\mathbb Q(\sqrt D)$. We estimate the size of the norm of an
indecomposable integer by expressing it as a power series in $u_i^{-1}$, where
$\sqrt D$ has the periodic continued fraction expansion $[u_0, u_1, u_2, \dots,
u_{s-1}, 2u_0, u_1, u_2, \dots]$. This enables us to disprove a conjecture of
Jang-Kim [JK] concerning the maximal size of the norm of an indecomposable
integer.Comment: 12 pages, to appear in J. Number Theor
We obtain good estimates on the ranks of universal quadratic forms over Shanks’ family of the simplest cubic fields and several other families of totally real number fields. As the main tool, we characterize all the indecomposable integers in these fields and the elements of the codifferent of small trace. We also determine the asymptotics of the number of principal ideals of norm less than the square root of the discriminant.
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