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arXiv · 1906.03705

Representation of integers by sparse binary forms

Shabnam Akhtari·Paloma Bengoechea
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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.

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Shabnam Akhtari, Paloma Bengoechea. 2019-06-09. Representation of integers by sparse binary forms. https://doi.org/10.1090/tran/8241

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