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 | References | Similar Articles | Additional Information

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.

**[1]***Handbook of mathematical functions, with formulas, graphs, and mathematical tables*, Edited by Milton Abramowitz and Irene A. Stegun, Dover Publications, Inc., New York, 1966. MR**0208797****[2]**Michael A. Bean,*Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula*, Trans. Amer. Math. Soc.**347**(1995), no. 12, 4959–4983. MR**1307999**, https://doi.org/10.1090/S0002-9947-1995-1307999-2**[3]**Michael A. Bean,*An isoperimetric inequality for the area of plane regions defined by binary forms*, Compositio Math.**92**(1994), no. 2, 115–131. MR**1283225****[4]**Michael A. Bean,*An isoperimetric inequality related to Thue’s equation*, Bull. Amer. Math. Soc. (N.S.)**31**(1994), no. 2, 204–207. MR**1260516**, https://doi.org/10.1090/S0273-0979-1994-00517-8**[5]**Michael A. Bean,*A note on the Thue inequality*, Proc. Amer. Math. Soc.**123**(1995), no. 7, 1975–1979. MR**1283540**, https://doi.org/10.1090/S0002-9939-1995-1283540-3**[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), no. 3-4, 207–247. MR**945012**, https://doi.org/10.1007/BF02392276**[10]**Wolfgang M. Schmidt,*Diophantine approximations and Diophantine equations*, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR**1176315****[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