Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Fuchs, Gesammelte Mathematische Werke, bd 1–3, Berlin, 1900–1906.
A. R. Forsyth, Theory of differential equations, v. I–VI, Dover reprint, 1959.
F.G. Frobenius, Gesammelte Abhandlungen, Springer, N.Y., 1968, Bd. 1.
H.T. Kung, J.F. Traub, All algebraic functions can be computed fast, J. Assoc. Comp. Mech. 25 (1978), 245–260.
I.L. Ince, Ordinary differential equations, Dover, 1959.
D.E. Kunth, The Art of Computer Programming, v. 2, 2nd ed., Addison-Wesley, 1981.
J. Della Dora, E. Tournier, Formal Solutions of Differential Equations in the Neighborhood of Singular Points, in SYMSAC' 81 Proceedings (ed. P.S. Wang), Assoc. Computer Machinery, 1981, pp. 25–30.
D.B. Chudnovsky, G.V. Chudnovsky, On expansion of algebraic functions in power and Puiseux series, IBM Research Report, RC11365, 9/13/85, 96 pp.
R.P. Brent, H.T. Kung, Fast algorithms for manipulating formal power series, J. Assoc. Comp. Machinery, 25 (1978), 581–595.
E. Hille, Ordinary differential equations in the complex domain, Wiley, N.Y., 1976.
B.C. Carlson, Special Functions of Applied Mathematics, Academic Press, N.Y., 1977.
R.P. Brent, Multiple-precision zero-finding method and the complexity of elementary function evaluation, in Analytic Computational Complexity (ed. by J.F. Traub), Academic Press, 1976, pp. 151–176.
D.V. Chudnovsky, G.V. Chudnovsky, Padé and rational approximations to systems of functions and their arithmetic applications, Lecture Notes Math., v. 1052, Springer, N.Y., 1984, 37–84.
D.V. Chudnovsky, G.V. Chudnovsky, Padé approximations to solutions of linear differential equations and applications to diophantine analysis, ibid., 85–167.
N.H. Abel, Sur l'intégration de la formule différentielle ρ·dx/√R, R and ρ êtant des fonctions entières, J. Reine Angew, Math., 1 (1826), 185–221, = Oeuvres, v. 1, 104–144.
F.G. Frobenius, L. Stickelberger, Über die addition und multiplication der elliptischen functionen, ibid., 88 (1880), 146–184.
H.F. Baker, Note on the foregoing paper "Commutative differential operators", by J.L. Burchnall and J.W. Chaundy, Proc. Roy. Soc. London, A., 118 (1928), 584–593.
G.V. Chudnovsky, Padé approximations and the Rieman monodromy problems, in Bifurcation Phenomena in Mathematical Physics, D. Reidel Publishing Company, Boston, 1980, 448–510.
D.V. Chudnovsky, Riemann monodromy problem, isomonodromy deformation equations and completely integrable systems, ibid., 385–447.
R.T. Baumel, J.L. Gammel, J. Nuttall, Asymptotic form of Hermite-Padé polynomials, IMA Journal of Appl. Math., 27 (1981), 335–357.
D.V. Chudnovsky, G.V. Chudnovsky, Sequences of numbers generated by addition in formal groups and new primality and factorization test, IBM Research Report, RC 11262, 7/12/85, 1985, 102 pp., Advances in Applied Mathematics, 1986 (to appear).
G. Andrews, Physics, Ramanujan and SCRATCHPAD, in Proceedings of the conference "Computer Algebra as a Tool of Research in Mathematics and Physics" (to appear).
Parallel Processing Systems (ed. by D. Evans), Cambridge Univ. Press, 1982.
H. vonKoch, Un théorem sur les intégrales irrégulieres des equations differentielles linéaires et son application du problème de l'integration, Ark. för Math., 13 (1918), No. 15, 1–18.
F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974.
H. Poincaré, Sur le déterminant de Hill, Bull. Astron., 17 (1900), 134–143 = Oeuvres, v. 8, 383–393, Paris, 1952.
M. Hazewinkel, Formal Groups and Applications, Academic Press, 1978.
E.R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, N.Y., 1973.
B. Riemann, Oeuvres Mathématiques, Blanchard, Paris, 1966, 353–363.
H. Poincaré, Sur les groupes des équations linéaires, Acta. Math., 5 (1884), 240–278.
J.A. Lappo-Danilevsky, Mémoires sur la Théorie des Systèmes des Équations Différentielles Linéaires, Chelsea, 1953.
The Riemann Problem, Complete Integrability and Arithmetic Applications, ed. by D.V. Chudnovsky and G.V. Chudnovsky, Lecture Notes in Math., v. 925, Springer, N.Y., 1982.
P. Deligne, Equations Differentielles à Point Singulier Reguliers, Lecture Notes in Math., v. 163, Springer, N.Y., 1970.
A. Hurwitz, Über Riemannshe flachen mit gegeben verzweigung-spunten, Math. Ann., 39 (1891), 1–61.
M. Fried, Fields of definition of function fields and Hurwitz families. Groups as Galois groups, Comm. Algebra, 5(1977), 17–82.
G.V. Belyi, On Galois extensions of a maximal cyclotomic field, field., Math. USSR Izvestija 14 (1980), 247–256.
B.N. Matzat, Konstruktion von zahl-and funktionen-körpern mit vorgegebener Galoisgruppe, J. Reine Angew. Math., 349 (1984), 179–220.
J.G. Thompson, Some finite groups which appear as Gal(L/K), where K ≤ θ(μn), J. Algebra, 89 (1984), 437–499.
B.H. Matzat, Zum einbettungsproblem der algebraischen zahlentheorie mit nicht abelschem kern, Invent Math., 80 (1985), 365–374.
G.V. Belyi, On extensions of the maximal cyclotomic field having a given classical Galois group, J. Reine Angew, Math., 341(1983), 147–156.
Proceedings of the Rutgers Group Theory Year, 1983–84 (Ed. by M. Aschbacher, D. Gorenstein, R. Lyons, M. O'Nan, C. Sims, W. Feit), Cambridge Univ. Press, 1984.
N.G. Chebotareff, Theory of Algebraic Functions, Gostechizdat, 1968 (Russian).
H.F. Baker, Abel's Theorem and the Allied Theory Including The Theory of Theta Functions, Cambridge, 1897.
J.H. Davenport, On the Integration of Algebraic Functions, Lecture Notes Computer Sci. v. 102, Springer, N.Y. (1981).
B. M. Trager, Integration of Algebraic Functions, Ph.D. Thesis, M.I.T. 1984.
D.V. Chudnovsky, G.V. Chudnovsky, Applications of Padé Approximations to the Grothendieck conjecture on linear differential equations, Lecture Notes in Math., Springer, v. 1135, 1985, N.Y., 52–100.
D.V. Chudnovsky, S.V. Chudnovsky, The Grothendieck conjecture and Padé approximations, Proc. Japan Acad., 61A (1985), 87–90.
F. Baldassarri, B.M. Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math. 101 (1970), 42–76.
D.V. Chudnovsky, G.V. Chudnovsky, p-adic properties of linear differential equations and Abelian integrals, IBM Research Report RC10645, 7/26/84.
S.V. Kowalewski, Über die reduction einer bestimintea classe Abel'schen integrale 3-en ranges auf elliptische integrale, Acta Math., 4 (1884), 393–414.
D.V. Chudnovsky, Meromorphic solutions of nonlinear partial differential equations and many particle completely integrable systems, J. Math. Phys., 20 (1970), 2416–2422.
D.V. Chudnovsky, G.V. Chudnovsky, Pade approximations and diophantine geometry, Proc. Natl. Acad. Sci. U.S.A. 82 (1985), 2212–2216.
J.-P. Serre, Quelques applications du théoreme de densité de Chebotarev, IHES Publ. Math., 54 (1981), 323–401.
G. Faltings, Endlichkeitssätze für abelsche varietäten über zahlkörpern, Inv. Math., 73 (1983), 349–366.
T. Honda, On the theory of commutative formal groups, J. Math. Soc. Japan, 22 (1970), 213–246.
M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math., 70 (1948), 31–74.
S. Lichtenbaum, On p-adic L-functions associated to elliptic curves, Inv. Math., 56 (1980), 19–55.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 1966.
P.S. Landweber, R.E. Stong, Circle actions on spin manifolds and characteristic numbers (to appear).
D.V. Chudnovsky, G.V. Chudnovsky, Elliptic modular functions and elliptic genera (to appear).
B. Dwork, Arithmetic theory of differential equations, Symposia Mathematica, v. 24, Academic Press, N.Y., 1981, 225–243.
D.V. Chudnovsky, G.V. Chudnovsky, Applications of Padé approximations to diophantine inequalities in values of G-functions, Lecture Notes in Math., Springer, v. 1135, 1985, pp. 9–51.
G. Shimura, On some problems of algebraicity,, Proceeding Intern. Congress of Math., Helsinki, 1978, v. 1, 373–379.
G. V Chudnovsky, Algebraic independence of values of exponential and elliptic functions, ibid., 339–355.
P. Deligne, Valeurs de fonctions L et périodes d'intégrales, Proc. Symp. Pure Math., v. 33, part 2, AMS, 1979, 313–346.
R. Fricke, F. Klein, Vorlesungen über die theorie der automorphen functionen, 2v., Teubner, 1926.
D.V. Chudnovsky, G.V. Chudnovsky, A random walk in higher arithmetic, Adv. Appl. Math., 7 (1986), 101–122.
S. Lang, Introduction to transcendental numbers, Addison-Wesley, 1966.
G.V. Chudnovsky, Explicit construction of auxiliary functions for transcendental numbers, Lecture Notes Math., v. 751, Springer, N.Y. 1979, 45–69.
H. Padé, Oeuvres, Blanchard, Paris, 1984.
N.P. Erugin, Lappo-Danilevsky method in the Theory of Differential Equations, Leningrad Univ. Press, 1956.
W. Magnus, Monodromy groups and Hill's equation, Comm. Pure Appl. Math., 29 (1976), 701–716.
F.M. Arscott, Periodic Differential Equations, Macmillan, N.Y., 1964.
L. Keen, H.E. Rauch, A.T. Vasques, Moduli of punctured tori and the accessory parameter of Lamé's equations, Bull. Amer. Math. Soc., 255 (1979), 201–230.
J.L. Ince, Periodic Lamé functions, Proc. Edinb. Math. Soc., 41 (1923), 94–100 and 60 (1939), 47–63.
P.L. Chebichef, Sur l'integration de la différentielle (X+A) dx/√x4+ax3+bx2+cx+d, Bull. Acad. Impériale de Saint-Pétersbourg, 3 (1861), 1–12.
E.G.C. Poole, Introduction to the Theory of Linear Differential Equations, Oxford, 1936.
J. Meixner, Orthogonal polynomials in the theory of Mathieu functions, I, II, Arch. Math. 36 (1981), 162–167; 39 (1982), 46–50.
L.A. Takhtadjan, P.G. Zograf, The Liouville equation action—the generating function for accessory parameters, Funct. Anal., 19 (1985), 67–68.
O. Perron, Die Lehre von den Kettenbrüchen, Teubner, 1929.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1987 Springer-Verlag
About this paper
Cite this paper
Chudnovsky, D.V., Chudnovsky, G.V. (1987). Computer assisted number theory with applications. In: Chudnovsky, D.V., Chudnovsky, G.V., Cohn, H., Nathanson, M.B. (eds) Number Theory. Lecture Notes in Mathematics, vol 1240. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072972
Download citation
DOI: https://doi.org/10.1007/BFb0072972
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-17669-5
Online ISBN: 978-3-540-47756-3
eBook Packages: Springer Book Archive