On expansion of algebraic functions in power and Puiseux series, I

Abstract We present algorithms that (a) reduce an algebraic equation, defining an algebraic function, to a Fuchsian differential equation that this function satisfies; and (b) compute coefficients in the expansions of solutions of linear differential equations in the neighborhood of regular singularities via explicit linear recurrences. This allows us to compute the Nth coefficient (or N coefficients) of an algebraic function of degree d in O(dN) operations with O(d) storage (or O(dN) storage).

[1]  G. V. Chudnovsky,et al.  Padé Approximation and the Riemann Monodromy Problem , 1980 .

[2]  Jan-Erik Björk,et al.  Rings of differential operators , 1979 .

[3]  H. T. Kung,et al.  Fast Algorithms for Manipulating Formal Power Series , 1978, JACM.

[4]  D. Chudnovsky,et al.  The Wronskian formalism for linear differential equations and Padé approximations , 1984 .

[5]  I. Kaplansky.  An introduction to differential algebra , 1957 .

[6]  Nils Nilsson,et al.  Asymptotic estimates for spectral functions connected with hypoelliptic differential operators , 1965 .

[7]  Georg Friedrich Bernhard Riemann.  Gesammelte mathematische Werke und wissenschaftlicher Nachlass , 1971 .

[8]  G. M.,et al.  Theory of Differential Equations , 1902, Nature.

[9]  D. V. Chudnovsky,et al.  On expansion of algebraic functions in power and Puiseux series, II , 1987, J. Complex..

[10]  D. T. Whiteside,et al.  The mathematical papers of Isaac Newton , 1967 .

[11]  Masaki Kashiwara,et al.  Seminar on micro-local analysis (AM-93) , 1980 .

[12]  D. V. Chudnovsky,et al.  Riemann Monodromy Problem, Isomonodromy Deformation Equations and Completely Integrable Systems , 1980 .

[13]  P. Deligne,et al.  Equations differentielles à points singuliers reguliers , 1970 .

[14]  R. J. Walker.  Algebraic curves , 1950 .

[15]  Nils Nilsson,et al.  Some growth and ramification properties of certain integrals on algebraic manifolds , 1964 .

[16]  H. T. Kung,et al.  All Algebraic Functions Can Be Computed Fast , 1978, JACM.

[17]  D V Chudnovsky,et al.  Padé approximations and diophantine geometry. , 1985, Proceedings of the National Academy of Sciences of the United States of America.

[18]  J. A. Lappo-Danilevsky.  Mémoires sur la théorie des systémes des équations différentielles linéaires , 1954 .

[19]  D. V. Chudnovsky,et al.  Applications of Padé approximations to the Grothendieck conjecture on linear differential equations , 1985 .

[20]  Richard P. Brent,et al.  On the Complexity of Composition and Generalized Composition of Power Series , 1980, SIAM J. Comput..

[21]  Gesammelte Abhandlungen , 1906, Nature.

[22]  G. V. Chudnovsky,et al.  Rational and Padé approximations to solutions of linear differential equations and the monodromy theory , 1980 .

[23]  M. S. Knebelman.  The Wronskian for Linear Differential Equations , 1949 .

[24]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[25]  D. Chudnovsky,et al.  Sequences of numbers generated by addition in formal groups and new primality and factorization tests , 1986 .

[26]  G. V. Chudnovsky,et al.  Number Theoretic Applications of Polynomials with Rational Coefficients Defined by Extremality Conditions , 1983 .