Bean, Michael A. Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula. (English) Zbl 0857.11014 Trans. Am. Math. Soc. 347, No. 12, 4959-4983 (1995). Summary: In a previous paper [Compos. Math. 92, 115-131 (1994; Zbl 0816.11026)] the author showed that if \(F\) is a binary form with complex coefficients having degree \(n \geq 3\) and discriminant \(D_F \neq 0\), and if \(A_F\) is the area of the region \(|F(x,y) |\leq 1\) in the real affine plane, then \(|D_F |^{1/n(n-1)} A_F \leq 3B ({1 \over 3}, {1 \over 3})\), where \(B({1 \over 3}, {1 \over 3})\) denotes the Beta function with arguments of \(1/3\). This inequality was derived by demonstrating that the sequence \(\{M_n\}\) defined by \(M_n=\max |D_F |^{1/n(n-1)} A_F\), where the maximum is taken over all forms of degree \(n\) with \(D_F \neq 0\), is decreasing, and then by showing that \(M_3=3B({1 \over 3}, {1 \over 3})\). The resulting estimate, \(A_F \leq 3B({1 \over 3}, {1 \over 3})\) 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 \(\{M_n\}\). By appealing to the theory of hypergeometric functions, it is shown that \(M_4=2^{7/6} B({1 \over 4}, {1 \over 2})\) and that \(M_4\) is attained for the form \(XY(X^2 - Y^2)\). It is also shown that there is a correspondence, arising from the Schwarz-Christoffel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that \(A_F\) is the perimeter of the polygon corresponding to \(F\). Based on this correspondence and a representation theorem for \(|D_F |^{1/n(n-1)} A_F\), it is conjectured that \(M_n=D_{F^*_n}^{1/n(n-1)} A_{F^*_n}\), where \(F^*_n(X,Y) = \prod^n_{k=1} (X \sin ({k \pi \over n})-Y \cos ({k \pi \over n}))\), and that the limiting value of the sequence \(\{M_n\}\) is \(2\pi\). Cited in 1 ReviewCited in 6 Documents MSC: 11D75 Diophantine inequalities 33C05 Classical hypergeometric functions, \({}_2F_1\) 11J25 Diophantine inequalities 30C20 Conformal mappings of special domains 51M25 Length, area and volume in real or complex geometry Keywords:hypergeometric functions; Schwarz-Christoffel mapping formula; binary forms; equiangular polygons Citations:Zbl 0816.11026 Software:Maple PDFBibTeX XMLCite \textit{M. A. Bean}, Trans. Am. Math. Soc. 347, No. 12, 4959--4983 (1995; Zbl 0857.11014) Full Text: DOI References: [1] M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York, 1965. · Zbl 0515.33001 [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. · Zbl 0816.11026 [3] Michael A. Bean, An isoperimetric inequality related to Thue’s equation, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 2, 204 – 207. · Zbl 0816.11025 [4] Michael A. Bean and Jeffrey Lin Thunder, Isoperimetric inequalities for volumes associated with decomposable forms, J. London Math. Soc. (2) 54 (1996), no. 1, 39 – 49. · Zbl 0854.11019 [5] E. Bombieri and W. M. Schmidt, On Thue’s equation, Invent. Math. 88 (1987), no. 1, 69 – 81. · Zbl 0614.10018 [6] B.W. Char et al., Maple library reference manual, Springer-Verlag, New York, 1991. · Zbl 0763.68046 [7] Ruel V. Churchill and James Ward Brown, Complex variables and applications, 4th ed., McGraw-Hill Book Co., New York, 1984. · Zbl 0546.30003 [8] E.T. Copson, An introduction to the theory of functions of a complex variable, Clarendon, Oxford, 1935. · Zbl 0012.16902 [9] L.E. Dickson, Algebraic invariants, Wiley, New York, 1914. · JFM 45.0196.10 [10] Christopher Hooley, On binary cubic forms, J. Reine Angew. Math. 226 (1967), 30 – 87. · Zbl 0163.04605 [11] Kurt Mahler, Zur Approximation algebraischer Zahlen. III, Acta Math. 62 (1933), no. 1, 91 – 166 (German). Über die mittlere Anzahl der Darstellungen grosser Zahlen durch binäre Formen. · Zbl 0008.19801 [12] J. Mueller and W. M. Schmidt, Thue’s equation and a conjecture of Siegel, Acta Math. 160 (1988), no. 3-4, 207 – 247. · Zbl 0655.10016 [13] Julia Mueller and W. M. Schmidt, On the Newton polygon, Monatsh. Math. 113 (1992), no. 1, 33 – 50. · Zbl 0770.12001 [14] G.C. Salmon, Modern higher algebra, 3rd ed., Dublin 1876, 4th ed., Dublin 1885 (reprinted New York, 1924). [15] Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. · Zbl 0754.11020 [16] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. [17] A. Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284-305. · JFM 40.0265.01 [18] E.T. Whittaker and G.N. Watson, Modern analysis, 4th ed., Cambridge 1927. · JFM 53.0180.04 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.