Algebraic solutions of the Lamé equation, revisited
Copyright policy )Algebraic solutions of the Lamé equation, revisited
A minor error in the necessary conditions for the algebraic form of the Lam�� equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lam��'s differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lam�� equation. [See R. C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric, Riemann, and Lam�� equations", J. Symbolic Computation 28 (1999), 521-545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.
20 pages, elsart document class, no figures
- University of Arizona United States
Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, hypergeometric equation, Projective monodromy group, Schwarz list, Lamé, Mathieu, and spheroidal wave functions, FOS: Physical sciences, Mathematical Physics (math-ph), algebraic solution, Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain, Algebraic solution, projective monodromy group, Hypergeometric equation, Mathematics - Classical Analysis and ODEs, Lamé equation, Classical Analysis and ODEs (math.CA), FOS: Mathematics, finite monodromy, Algebraic functions and function fields in algebraic geometry, 34A20 (Primary) 33E10,14H05 (Secondary), Analysis, Mathematical Physics, Finite monodromy
Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, hypergeometric equation, Projective monodromy group, Schwarz list, Lamé, Mathieu, and spheroidal wave functions, FOS: Physical sciences, Mathematical Physics (math-ph), algebraic solution, Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain, Algebraic solution, projective monodromy group, Hypergeometric equation, Mathematics - Classical Analysis and ODEs, Lamé equation, Classical Analysis and ODEs (math.CA), FOS: Mathematics, finite monodromy, Algebraic functions and function fields in algebraic geometry, 34A20 (Primary) 33E10,14H05 (Secondary), Analysis, Mathematical Physics, Finite monodromy
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