Representation of integers by sparse binary forms

Akhtari, Shabnam; Bengoechea, Paloma

doi:10.1090/tran/8241

Representation of integers by sparse binary forms
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by Shabnam Akhtari and Paloma Bengoechea PDF

Trans. Amer. Math. Soc. 374 (2021), 1687-1709 Request permission

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 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 [Trans. Amer. Math. Soc. 303 (1987), pp. 241–255], [Acta Math. 160 (1988), pp. 207–247], in special but important cases.

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