Power series solutions of singular (q)-differential equations

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyse the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.

[1]  O. Ore.  Theory of Non-Commutative Polynomials , 1933 .

[2]  Éric Schost,et al.  Fast computation of power series solutions of systems of differential equations , 2006, SODA '07.

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

[4]  Mark Giesbrecht,et al.  Factoring in Skew-Polynomial Rings over Finite Fields , 1998, J. Symb. Comput..

[5]  Michael J. Fischer,et al.  Fast on-line integer multiplication , 1973, STOC '73.

[6]  A. J. Stothers.  On the complexity of matrix multiplication , 2010 .

[7]  Gilles Villard,et al.  Computing the rank and a small nullspace basis of a polynomial matrix , 2005, ISSAC.

[8]  E. G. C. Poole,et al.  Introduction to the theory of linear differential equations , 1936 .

[9]  H. Q. Le.  A direct algorithm to construct the minimal Z-pairs for rational functions , 2003, Adv. Appl. Math..

[10]  Joris van der Hoeven.  FFT-like Multiplication of Linear Differential Operators , 2002, J. Symb. Comput..

[11]  Ziming Li,et al.  A subresultant theory for Ore polynomials with applications , 1998, ISSAC '98.

[12]  Arne Storjohann,et al.  High-order lifting and integrality certification , 2003, J. Symb. Comput..

[13]  Dima Grigoriev,et al.  Complexity of Factoring and Calculating the GCD of Linear Ordinary Differential Operators , 1990, J. Symb. Comput..

[14]  Éric Schost,et al.  Differential equations for algebraic functions , 2007, ISSAC '07.

[15]  Joris van der Hoeven,et al.  Relax, but Don't be Too Lazy , 2002, J. Symb. Comput..

[16]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[17]  Alin Bostan,et al.  Algorithmique efficace pour des opérations de base en calcul formel. (Fast algorithms for basic operations in computer algebra) , 2003 .

[18]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[19]  Christopher Umans,et al.  Fast Polynomial Factorization and Modular Composition , 2011, SIAM J. Comput..

[20]  Mark Giesbrecht,et al.  Factoring and decomposing ore polynomials over Fq(t) , 2003, ISSAC '03.

[21]  F. Fauvet,et al.  Remarques algorithmiques liées au rang d’un opérateur différentiel linéaire , 2003 .

[22]  Öystein Ore.  Formale Theorie der linearen Differentialgleichungen. (Erster Teil). , 1932 .

[23]  Nicolas Le Roux,et al.  Products of ordinary differential operators by evaluation and interpolation , 2008, ISSAC '08.

[24]  Abraham Waksman.  On Winograd's Algorithm for Inner Products , 1970, IEEE Transactions on Computers.

[25]  Éric Schost,et al.  Optimization techniques for small matrix multiplication , 2011, ACCA.

[26]  Moulay A. Barkatou,et al.  A Monomial-by-Monomial Method for Computing Regular Solutions of Systems of Pseudo-Linear Equations , 2010, Math. Comput. Sci..

[27]  Henry Blumberg,et al.  Über algebraische Eigenschaften von linearen homogenen Differentialausdrücken , 1912 .

[28]  Daniel J. Bernstein.  Composing Power Series Over a Finite Ring in Essentially Linear Time , 1998, J. Symb. Comput..

[29]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[30]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[31]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[32]  Reynald Lercier,et al.  On Elkies subgroups of ‘ -torsion points in elliptic curves defined over a finite field , 2009 .

[33]  Éric Schost,et al.  Fast algorithms for differential equations in positive characteristic , 2009, ISSAC '09.

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

[35]  Éric Schost,et al.  Fast algorithms for computing isogenies between elliptic curves , 2006, Math. Comput..

[36]  Arnold Schönhage,et al.  Schnelle Multiplikation großer Zahlen , 1971, Computing.

[37]  Joris van der Hoeven.  Relaxed resolution of implicit equations , 2009 .

[38]  Arne Storjohann.  Notes on computing minimal approximant bases , 2006, Challenges in Symbolic Computation Software.

[39]  Éric Schost,et al.  Power series composition and change of basis , 2008, ISSAC '08.

[40]  Mark van Hoeij,et al.  A modular algorithm for computing the exponential solutions of a linear differential operator , 2004, J. Symb. Comput..

[41]  Peter Kirrinnis,et al.  Fast algorithms for the Sylvester equation AX-XBT=C , 2001, Theor. Comput. Sci..

[42]  Werner Balser,et al.  Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations , 1999 .

[43]  Moulay A. Barkatou,et al.  An Algorithm Computing the Regular Formal Solutions of a System of Linear Differential Equations , 1999, J. Symb. Comput..

[44]  Richard P. Stanley,et al.  Differentiably Finite Power Series , 1980, Eur. J. Comb..

[45]  Venkatesan Guruswami,et al.  Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy , 2005, IEEE Transactions on Information Theory.

[46]  Erich Kaltofen,et al.  On fast multiplication of polynomials over arbitrary algebras , 1991, Acta Informatica.

[47]  Joris van der Hoeven.  Relaxed mltiplication using the middle product , 2003, ISSAC.

[48]  H. Q. Le,et al.  Univariate Ore Polynomial Rings in Computer Algebra , 2005 .

[49]  Joris van der Hoeven.  New algorithms for relaxed multiplication , 2007, J. Symb. Comput..

[50]  Mark Giesbrecht,et al.  Factoring in Skew-Polynomial Rings , 1992, LATIN.

[51]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[52]  Éric Schost,et al.  Polynomial evaluation and interpolation on special sets of points , 2005, J. Complex..

[53]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[54]  J. H. M. Wedderburn,et al.  Non-commutative domains of integrity. , 1932 .