Binary Forms, Hypergeometric Functions and the Schwarz-christoffel Mapping Formula

In a previous paper, it was shown that HE is a binary form with complex coefficients having degree n > 3 and discriminant Df ^ 0, and if Af is the area of the region \F(x, y)\ < 1 in the real affine plane, then \DF\llni-"-^AF < 1B{\, \), where B{\ , 1) denotes the Beta function with arguments of 1/3 . This inequality was derived by demonstrating that the sequence {M„} defined by Mn = max|Df \Uif.»-^)AF, where the maximum is taken over all forms of degree n with Df ^ 0, is decreasing, and then by showing that M-¡ = 7>B{\, \). The resulting estimate, Af < 35(j, \) 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 {Mi} • By appealing to the theory of hypergeometric functions, it is shown that Mi, = 21leB(\, \) and that M<¡, is attained for the form XY{X2 Y2). 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 Af is the perimeter of the polygon corresponding to F . Based on this correspondence and a representation theorem for \Df\1fn(-"~l^Af , it is conjectured that Mn = DlF'?{n-l)Af. .where F„'(X, Y) = Wk=i(x ^(f) ~ Yca$(**)), and that the limiting value of the sequence {Mn} is 2n . ■