Complexity of creative telescoping for bivariate rational functions

The long-term goal initiated in this work is to obtain fast algorithms and implementations for definite integration in Almkvist and Zeilberger's framework of (differential) creative telescoping. Our complexity-driven approach is to obtain tight degree bounds on the various expressions involved in the method. To make the problem more tractable, we restrict to bivariate rational functions. By considering this constrained class of inputs, we are able to blend the general method of creative telescoping with the well-known Hermite reduction. We then use our new method to compute diagonals of rational power series arising from combinatorics.

[1]  Ch. Hermite.  Sur l'intégration des fractions rationnelles , 1872 .

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

[3]  L. Lipshitz,et al.  The diagonal of a D-finite power series is D-finite , 1988 .

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

[5]  Keith O. Geddes,et al.  AN ALGORITHM TO COMPUTE THE MINIMAL TELESCOPERS FOR RATIONAL FUNCTIONS (DIFFERENTIAL – INTEGRAL CASE) , 2002 .

[6]  Doron Zeilberger,et al.  The Method of Differentiating under the Integral Sign , 1990, J. Symb. Comput..

[7]  Doron Zeilberger,et al.  Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory , 2006, Adv. Appl. Math..

[8]  J. Urgen Gerhard.  Fast Modular Algorithms for Squarefree Factorization and Hermite Integration , 1999 .

[9]  L. Lipshitz,et al.  D-finite power series , 1989 .

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

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

[12]  E. Horowitz.  Algorithms for partial fraction decomposition and rational function integration , 1971, SYMSAC '71.

[13]  Jürgen Gerhard,et al.  Modular Algorithms in Symbolic Summation and Symbolic Integration , 2005, Lecture Notes in Computer Science.

[14]  R. Risch.  The problem of integration in finite terms , 1969 .

[15]  M. H. Protter,et al.  THE SOLUTION OF THE PROBLEM OF INTEGRATION IN FINITE TERMS , 1970 .

[16]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[17]  Abraham D. Flaxman,et al.  Strings with Maximally Many Distinct Subsequences and Substrings , 2004, Electron. J. Comb..

[18]  D. Zeilberger.  A holonomic systems approach to special functions identities , 1990 .

[19]  H. Q. Le.  ON THE DIFFERENTIAL-INTEGRAL ANALOGUE OF ZEILBERGER'S ALGORITHM TO RATIONAL FUNCTIONS , 2000 .

[20]  Manuel Bronstein.  Symbolic Integration I: Transcendental Functions (Algorithms and Computation in Mathematics) , 2004 .

[21]  Doron Zeilberger,et al.  The Method of Creative Telescoping , 1991, J. Symb. Comput..