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Computing Approximate Greatest Common Right Divisors of Differential Polynomials

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Abstract

Differential (Ore) type polynomials with “approximate” polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate greatest common right divisor problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given two differential polynomials, we present an algorithm to find nearby differential polynomials with a non-trivial GCRD, where nearby is defined with respect to a suitable coefficient norm. Intuitively, given two linear differential polynomials as input, the (approximate) GCRD problem corresponds to finding the (approximate) differential polynomial whose solution space is the intersection of the solution spaces of the two inputs. The approximate GCRD problem is proven to be locally well posed. A method based on the singular value decomposition of a differential Sylvester matrix is developed to produce an initial approximation of the GCRD. With a sufficiently good initial approximation, Newton iteration is shown to converge quadratically to an optimal solution. Finally, sufficient conditions for existence of a solution to the global problem are presented along with examples demonstrating that no solution exists when these conditions are not satisfied.

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Notes

  1. The polynomial coefficients of \(\partial ^i\) have the same degree, i.e., \(\deg {\widetilde{f}}_i \le \deg f_i\) and \(\deg {\widetilde{g}}_i \le g_i\).

  2. The inflated differential Sylvester matrix has more columns than rows; however, the nullspace of the columns contains the information pertaining to the GCRD. The trivial singular values are the zero singular values occurring from there being more columns than rows.

  3. A proof-of-concept implementation of the algorithms is available at https://www.scg.uwaterloo.ca/software/ApproxOreFoCM-2019.tgz.

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Acknowledgements

The authors would like to thank George Labahn for his comments. The authors would also like to thank the two anonymous referees for their careful reading and comments.

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Authors and Affiliations

  1. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

    Mark Giesbrecht & Joseph Haraldson

  2. Department of Mathematics, North Carolina State University, Raleigh, USA

    Erich Kaltofen

Authors
  1. Mark Giesbrecht
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  2. Joseph Haraldson
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  3. Erich Kaltofen
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Correspondence to Mark Giesbrecht.

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This research was partly supported by the Natural Sciences and Engineering Research Council (NSERC) Canada (Giesbrecht and Haraldson) and by the National Science Foundation (NFS) under Grant CCF-1421128 (Kaltofen).

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Giesbrecht, M., Haraldson, J. & Kaltofen, E. Computing Approximate Greatest Common Right Divisors of Differential Polynomials. Found Comput Math 20, 331–366 (2020). https://doi.org/10.1007/s10208-019-09422-2

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