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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
DOI: https://doi.org/10.1090/S0025-5718-97-00815-6
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.


References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1966. MR 34:8606
  • [2] M. A. Bean, Binary forms, hypergeometric functions, and the Schwarz-Christoffel mapping formula, Trans. Amer. Math. Soc. 347 (12) (1995), 4959-4983. MR 96c:11038
  • [3] -, An isoperimetric inequality for the area of plane regions defined by binary forms, Compositio Math. 92 (2) (1994), 115-131. MR 95i:11078
  • [4] -, An isoperimetric inequality related to Thue's equation, Bull. Amer. Math. Soc. 31 (2) (1994), 204-207. MR 95b:11034
  • [5] -, A note on the Thue inequality, Proc. Amer. Math. Soc. 123 (7) (1995), 1975-1979. MR 95i:11079
  • [6] B. W. Char et al. Maple V Library Reference Manual, Springer-Verlag, New York, 1991.
  • [7] R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Wadsworth & Brooks-Cole, 1990. MR 92b:32001a,b,c
  • [8] K. Mahler, Zur Approximation algebraischer Zahlen III, Acta Math. 62 (1934), 91-166.
  • [9] J. Mueller and W. M. Schmidt, Thue's equation and a conjecture of Siegel, Acta Math. 160 (1988), 207-247. MR 89g:11029
  • [10] W. M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Math., vol. 1467, Springer-Verlag, New York, 1991. MR 94f:11059
  • [11] A. Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284-305.
  • [12] S. Wolfram, Mathematica: a system for doing mathematics by computer, 2nd ed., Addison-Wesley, 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: https://doi.org/10.1090/S0025-5718-97-00815-6
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

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