## The practical computation of areas associated with binary quartic forms

HTML articles powered by AMS MathViewer

- by Michael A. Bean PDF
- Math. Comp.
**66**(1997), 1269-1293 Request permission

## 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

- Milton Abramowitz and Irene A. Stegun (eds.),
*Handbook of mathematical functions, with formulas, graphs, and mathematical tables*, Dover Publications, Inc., New York, 1966. MR**0208797** - 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**, DOI 10.1090/S0002-9947-1995-1307999-2 - 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** - 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**, DOI 10.1090/S0273-0979-1994-00517-8 - Michael A. Bean,
*A note on the Thue inequality*, Proc. Amer. Math. Soc.**123**(1995), no. 7, 1975–1979. MR**1283540**, DOI 10.1090/S0002-9939-1995-1283540-3 - B. W. Char et al.
*Maple V Library Reference Manual*, Springer-Verlag, New York, 1991. - W. J. Trjitzinsky,
*General theory of singular integral equations with real kernels*, Trans. Amer. Math. Soc.**46**(1939), 202–279. MR**92**, DOI 10.1090/S0002-9947-1939-0000092-6 - K. Mahler,
*Zur Approximation algebraischer Zahlen III*, Acta Math.**62**(1934), 91–166. - 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**, DOI 10.1007/BF02392276 - Wolfgang M. Schmidt,
*Diophantine approximations and Diophantine equations*, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR**1176315**, DOI 10.1007/BFb0098246 - A. Thue,
*Über Annäherungswerte algebraischer Zahlen*, J. Reine Angew. Math.**135**(1909), 284–305. - S. Wolfram,
*Mathematica: a system for doing mathematics by computer*, 2nd ed., Addison-Wesley, New York, 1991.

## Additional Information

**Michael A. Bean**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Email: mbean@math.lsa.umich.edu
- Received by editor(s): August 2, 1994
- Received by editor(s) in revised form: February 14, 1996
- © Copyright 1997 American Mathematical Society
- 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