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Mathematics of Computation

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The practical computation of areas associated with binary quartic forms

Author: Michael A. Bean
Journal: Math. Comp. 66 (1997), 1269-1293
MSC (1991): Primary 11D75, 51M25; Secondary 11-04, 11E76, 11H06, 33C05, 51M16
MathSciNet review: 1397439
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Abstract: We derive formulas for practically computing the area of the region $|F(x,y)| \leq 1$ defined by a binary quartic form $F(X,Y) \in \mathbb R [X,Y]$. These formulas, which involve a particular hypergeometric function, are useful when estimating the number of lattice points in certain regions of the type $|F(x,y)| \leq h$ and will likely find application in many contexts. We also show that for forms $F$ of arbitrary degree, the maximal size of the area of the region $|F(x,y)| \leq 1$, normalized with respect to the discriminant of $F$ and taken with respect to the number of conjugate pairs of $F(x,1)$, increases as the number of conjugate pairs decreases; and we give explicit numerical values for these normalized maxima when $F$ is a quartic form.

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Additional Information

Michael A. Bean
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Keywords: Elliptic integral, hypergeometric function, Thue inequality
Received by editor(s): August 2, 1994
Received by editor(s) in revised form: February 14, 1996
Article copyright: © Copyright 1997 American Mathematical Society