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Journal Article

Representation of integers by sparse binary forms

MPS-Authors
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Akhtari,  Shabnam
Max Planck Institute for Mathematics, Max Planck Society;

External Ressource

https://doi.org/10.1090/tran/8241
(Publisher version)

Fulltext (public)

arXiv:1906.03705.pdf
(Preprint), 294KB

Supplementary Material (public)
There is no public supplementary material available
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.

Cite as: http://hdl.handle.net/21.11116/0000-0007-ABE5-6
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.