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Journal Article

Representation of integers by sparse binary forms

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Akhtari, Shabnam
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Akhtari, S., & Bengoechea, P. (in press). Representation of integers by sparse binary forms. Transactions of the American Mathematical Society, Published Online - Print pending. doi:10.1090/tran/8241.

Abstract

We will give new upper bounds for the number of solutions to the inequalities
of the shape $|F(x , y)| \leq h$, where $F(x , y)$ is a sparse binary form,
with integer coefficients, and $h$ is a sufficiently small integer in terms of
the absolute value of the discriminant of the binary form $F$. Our bounds
depend on the number of non-vanishing coefficients of $F(x , y)$. When $F$ is
really sparse, we establish a sharp upper bound for the number of solutions
that is linear in terms of the number of non-vanishing coefficients. This work
will provide affirmative answers to a number of conjectures posed by Mueller
and Schmidt in 1988, for special but important cases.