The practical computation of areas associated with binary quartic forms
Author:
Michael A. Bean
Journal:
Math. Comp. 66 (1997), 12691293
MSC (1991):
Primary 11D75, 51M25; Secondary 1104, 11E76, 11H06, 33C05, 51M16
MathSciNet review:
1397439
Fulltext PDF Free Access
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Abstract: We derive formulas for practically computing the area of the region defined by a binary quartic form . These formulas, which involve a particular hypergeometric function, are useful when estimating the number of lattice points in certain regions of the type and will likely find application in many contexts. We also show that for forms of arbitrary degree, the maximal size of the area of the region , normalized with respect to the discriminant of and taken with respect to the number of conjugate pairs of , increases as the number of conjugate pairs decreases; and we give explicit numerical values for these normalized maxima when is a quartic form.
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 [2]
 M. A. Bean, Binary forms, hypergeometric functions, and the SchwarzChristoffel mapping formula, Trans. Amer. Math. Soc. 347 (12) (1995), 49594983. MR 96c:11038
 [3]
 , An isoperimetric inequality for the area of plane regions defined by binary forms, Compositio Math. 92 (2) (1994), 115131. MR 95i:11078
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 , An isoperimetric inequality related to Thue's equation, Bull. Amer. Math. Soc. 31 (2) (1994), 204207. MR 95b:11034
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 , A note on the Thue inequality, Proc. Amer. Math. Soc. 123 (7) (1995), 19751979. MR 95i:11079
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 B. W. Char et al. Maple V Library Reference Manual, SpringerVerlag, New York, 1991.
 [7]
 R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Wadsworth & BrooksCole, 1990. MR 92b:32001a,b,c
 [8]
 K. Mahler, Zur Approximation algebraischer Zahlen III, Acta Math. 62 (1934), 91166.
 [9]
 J. Mueller and W. M. Schmidt, Thue's equation and a conjecture of Siegel, Acta Math. 160 (1988), 207247. MR 89g:11029
 [10]
 W. M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Math., vol. 1467, SpringerVerlag, New York, 1991. MR 94f:11059
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 A. Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284305.
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 S. Wolfram, Mathematica: a system for doing mathematics by computer, 2nd ed., AddisonWesley, New York, 1991.
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Additional Information
Michael A. Bean
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
mbean@math.lsa.umich.edu
DOI:
http://dx.doi.org/10.1090/S0025571897008156
PII:
S 00255718(97)008156
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
