Bavula*

   A Maple package for linear ordinary polynomial integro-differential systems

   
* The name of the package refers to the following article, which contains the original ideas that motivated our work:

V. V. Bavula, The algebra of integro-differential operators on an affine line and its modules. Journal of Pure and Applied Algebra, 217 (3), 495–529, 2013.

 

Authors: T. Cluzeau, C. Pinto, A. Quadrat

I - Content

This web page is dedicated to a Maple package called Bavula which contains implementations of the algorithms developed in the papers

 

[1] Cluzeau, T., Pinto, C., Quadrat, A.,. An algorithmic proof of the coherence of the ring of polynomial ordinary integro-differential operators. Submitted to ISSAC 2025.

[2] Cluzeau, T., Pinto, C., Quadrat, A.,. Polynomial solutions for general linear polynomial ordinary integro-differential systems. Submitted to ISSAC 2025.

[3] Cluzeau, T., Pinto, C., Quadrat, A., The effective Cramer property of the ring of polynomial ordinary integro-differential operators. In preparation, 2025.


II - Download and installation

Our package is built upon three other Maple packages that you need to download if you want to use the Bavula package.

 

1.     IntDiffOp: Bavula uses the arithmetic of integro-differential operators implemented in the Maple package IntDiffOp developed by A. Korporal, G. Regensburger, and M. Rosenkranz: you can download it at http://gregensburger.com/software/IntDiffOp.zip 

2.     OreModules for computing with matrices with entries in the first Weyl algebra of linear ordinary differential operators with polynomial coefficients and computing in the skew Laurent polynomial ring B_1=I_1/<e> (see [2,3])

3.     IntegrableConnections for computing polynomial solutions of integrable connections

 

 

Our package is available for download: Bavula.mla


To install it, you must proceed as follows:

  1. Copy the previous .mla file
  2. Open it with Maple (Execute march(‘open’, "…./Bavula.mla");)

  3. Type

    with(Bavula);

    You must get the list of the functions contained in the package. If you do obtain an error message, then you have probably done something wrong in Step 2.

If you do not manage to install the package, then contact us.


III - Examples of calculations

  1. Polynomial Solutions (PolySolsI1)

    Input: - a q * p matrix P with entries in the ring of ordinary polynomial integro-differential operators
               - a vector w of size q with entries in the ring of ordinary polynomial integro-differential operators

    Output: a parametrization of all the polynomial solutions v of the inhomogeneous linear system of polynomial ordinary integro-differential equations P(v)=w.

    Example file: ComputationOfPolynomialSolutions.mw ComputationOfPolynomialSolutions.pdf
  2. Left/right Syzygies (SyzygyI1)

    Input: a matrix P with entries in the ring of ordinary polynomial integro-differential operators
          
    Output: a matrix Q with entries in the ring of ordinary polynomial integro-differential operators such that the rows of Q generate the left sysygies (kernel) of P, i.e., ker(.P)=im(.Q).

    Example file: ComputationOfSyzygies.mw ComputationOfSyzygies.pdf
  3. Left/right inverses (LeftInverseI1)

    Input: a q * p matrix P with entries in the ring of ordinary polynomial integro-differential operators
          
    Output: a p * q matrix L such that L*P=I_p if it exists, [ ] if it does not exist.

    Example file: ComputationOfInverses.mw  ComputationOfInverses.pdf
  4. Left/right factorizations (FactorizeI1)

    Input: two matrices P and Q with entries in the ring of ordinary polynomial integro-differential operators,
          
    Output: a matrix F such that P = F*Q if it exists, [ ] if it does not exist.

    Example file: ComputationOfFactorizations.mw ComputationOfFactorizations.pdf



IV - Bug reports


If you have any problem with the package, find a bug or want to ask questions, then contact us.