Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula

Author:
Michael A. Bean

Journal:
Trans. Amer. Math. Soc. **347** (1995), 4959-4983

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 Schwarz-Christoifel 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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DOI:
https://doi.org/10.1090/S0002-9947-1995-1307999-2

Article copyright:
© Copyright 1995
American Mathematical Society