TY - JOUR

T1 - Representation of integers by sparse binary forms

AU - AKHTARI, SHABNAM

AU - BENGOECHEA, PALOMA

N1 - Publisher Copyright:
© 2020 American Mathematical Society.

PY - 2021/3

Y1 - 2021/3

N2 - We will give new upper bounds for the number of solutions to the inequalities of the shape |F(x, y)| ≤ 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.

AB - We will give new upper bounds for the number of solutions to the inequalities of the shape |F(x, y)| ≤ 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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U2 - 10.1090/tran/8241

DO - 10.1090/tran/8241

M3 - Article

AN - SCOPUS:85101627005

SN - 0002-9947

VL - 374

SP - 1687

EP - 1709

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 3

ER -