Groebner Bases Applied to Systems of Linear Difference Equations

In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high energy physics as recurrence relations for multiloop Feynman integrals. The most universal algorithmic tool for investigation of linear difference systems is based on their transformation into an equivalent Groebner basis form. We present an algorithm for this transformation implemented in Maple. The algorithm and its implementation can be applied to automatic generation of difference schemes for linear partial differential equations and to reduction of Feynman integrals. Some illustrative examples are given.

[1]  An algorithm to construct Gröbner bases for solving integration by parts relations , 2006, hep-ph/0602078.

[2]  Applying Gröbner bases to solve reduction problems for Feynman integrals , 2005, hep-lat/0509187.

[3]  S-bases as a tool to solve reduction problems for Feynman integrals , 2006, hep-ph/0606247.

[4]  F. B. Hildebrand.  Finite-difference equations and simulations , 1968 .

[5]  Joachim von zur Gathen,et al.  Modern Computer Algebra (3. ed.) , 2003 .

[6]  A. V. Mikhalev,et al.  Differential and Difference Dimension Polynomials , 1998 .

[7]  Y. Wen,et al.  Solving Linear Difference Systems with Lagged Expectations by a Method of Undetermined Coefficients , 2006 .

[8]  Computation of Gröbner bases for two-loop propagator type integrals , 2004, hep-ph/0403253.

[9]  N. Bose.  Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory , 1995 .

[10]  Maurice Janet,et al.  Leçons sur les systèmes d'équations aux dérivées partielles , 1930, Nature.

[11]  Vladimir P. Gerdt,et al.  Involutive Algorithms for Computing Groebner Bases , 2005 .

[12]  G. Hooft,et al.  Regularization and Renormalization of Gauge Fields , 1972 .

[13]  Vladimir P. Gerdt,et al.  Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations , 2006, ArXiv.

[15]  A. Peterson,et al.  Difference Equations: An Introduction with Applications , 2000 .

[16]  V. Smirnov.  Evaluating Feynman Integrals , 2005 .

[17]  A. Grozin.  Lectures on multiloop calculations , 2003, hep-ph/0307297.

[18]  B. Buchberger,et al.  Gröbner bases and applications , 1998 .

[19]  F. Tkachov.  A theorem on analytical calculability of 4-loop renormalization group functions , 1981 .

[20]  Vladimir P. Gerdt,et al.  A Maple Package for Computing Groebner Bases for Linear Recurrence Relations , 2005, ArXiv.

[21]  Vladimir P. Gerdt.  On computation of Gröbner bases for linear difference systems , 2006 .

[22]  Yuri A. Blinkov,et al.  Involutive bases of polynomial ideals , 1998, math/9912027.

[23]  Vladimir P. Gerdt,et al.  Janet-Like Monomial Division , 2005, CASC.

[24]  F. Tkachov,et al.  Integration by parts: The algorithm to calculate β-functions in 4 loops , 1981 .