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Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

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The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized and differential Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. Berman and Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

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Author information

Author notes

  1. Andrei Minchenko

    Present address: Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, 118 route de Narbonne, Toulouse Cedex 9, 31062, France

    Charlotte Hardouin

  2. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 234 Herzl Street, Rehovot, 7610001, Israel

    Andrei Minchenko

  3. Department of Mathematics, CUNY Queens College, 65-30 Kissena Blvd, Queens, NY, 11367, USA

    Alexey Ovchinnikov

  4. Department of Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, NY, 10016, USA

    Alexey Ovchinnikov

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  1. Charlotte Hardouin
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  2. Andrei Minchenko
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  3. Alexey Ovchinnikov
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Corresponding author

Correspondence to Alexey Ovchinnikov.

Additional information

Communicated by Nalini Anantharaman.

This work was partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02, by ANR-10-JCJC 0105, by ANR JR-13-JS01-0002-0, by the NSF grants CCF-0952591 and DMS-1413859, by the NSA grant H98230-15-1-0245, by the ISF grant 756/12, and by the Minerva foundation with funding from the Federal German Ministry for Education and Research.

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Hardouin, C., Minchenko, A. & Ovchinnikov, A. Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence. Math. Ann. 368, 587–632 (2017). https://doi.org/10.1007/s00208-016-1442-x

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