Binary forms, hypergeometric functions and the SchwarzChristoffel mapping formula
Author:
Michael A. Bean
Journal:
Trans. Amer. Math. Soc. 347 (1995), 49594983
MSC:
Primary 11D75; Secondary 11J25, 33C05
MathSciNet review:
1307999
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Abstract: In a previous paper, it was shown that if is a binary form with complex coefficients having degree and discriminant , and if is the area of the region in the real affine plane, then , where denotes the Beta function with arguments of 1/3. This inequality was derived by demonstrating that the sequence defined by , where the maximum is taken over all forms of degree with , is decreasing, and then by showing that . The resulting estimate, for such forms with integer coefficients, has had significant consequences for the enumeration of solutions of Thue inequalities. This paper examines the related problem of determining precise values for the sequence . By appealing to the theory of hypergeometric functions, it is shown that and that is attained for the form . It is also shown that there is a correspondence, arising from the SchwarzChristoifel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that is the perimeter of the polygon corresponding to . Based on this correspondence and a representation theorem for , it is conjectured that , where , and that the limiting value of the sequence is .
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 [1]
 M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York, 1965.
 [2]
 M.A. Bean, An isoperimetric inequality for the area of plane regions defined by binary forms, Compositio Math. 92 (1994), 115131. MR 1283225 (95i:11078)
 [3]
 , An isoperimetric inequality related to Thue's equation, Bull. Amer. Math. Soc. 31 (1994), 204207. MR 1260516 (95b:11034)
 [4]
 M.A. Bean and J.L. Thunder, Isoperimetric inequalities for volumes associated with decomposable forms, J. London Math. Soc. (to appear). MR 1395066 (97b:11087)
 [5]
 E. Bombieri and W.M. Schmidt, On Thue's equation, Invent. Math. 88 (1987), 6981. MR 877007 (88d:11026)
 [6]
 B.W. Char et al., Maple library reference manual, SpringerVerlag, New York, 1991.
 [7]
 R.V. Churchill and J.W. Brown, Complex variables and applications, 5th ed., McGrawHill, New York, 1990. MR 730937 (84k:30002)
 [8]
 E.T. Copson, An introduction to the theory of functions of a complex variable, Clarendon, Oxford, 1935.
 [9]
 L.E. Dickson, Algebraic invariants, Wiley, New York, 1914.
 [10]
 C. Hooley, On binary cubic forms, J. Reine Angew. Math. 226 (1967), 3087. MR 0213299 (35:4163)
 [11]
 K. Mahler, Zur Approximation algebraischer Zahlen III, Acta Math. 62 (1934), 91166. MR 1555382
 [12]
 J. Mueller and W.M. Schmidt, Thue's equation and a conjecture of Siegel, Acta Math. 160 (1988), 207247. MR 945012 (89g:11029)
 [13]
 , On the Newton polygon, Monatsh. Math. 113 (1992), 3350. MR 1149059 (93f:12003)
 [14]
 G.C. Salmon, Modern higher algebra, 3rd ed., Dublin 1876, 4th ed., Dublin 1885 (reprinted New York, 1924).
 [15]
 W.M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Math., vol. 1467, SpringerVerlag, New York, 1991. MR 1176315 (94f:11059)
 [16]
 L.J. Slater, Generalized hypergeometric functions, Cambridge, 1966. MR 0201688 (34:1570)
 [17]
 A. Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284305.
 [18]
 E.T. Whittaker and G.N. Watson, Modern analysis, 4th ed., Cambridge 1927.
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DOI:
http://dx.doi.org/10.1090/S00029947199513079992
PII:
S 00029947(1995)13079992
Article copyright:
© Copyright 1995 American Mathematical Society
