We establish some comparison results among the different parameterized Galois theories for -difference equations, completing the work [4], that addresses the problem in the case without parameters. Our main result is the link between the abstract parameterized Galois theories, that give information on the differential properties of abstract solutions of -difference equations, and the properties of meromorphic solutions of such equations. Notice that a linear -difference equation with meromorphic coefficients always admits a basis of meromorphic solutions, as proven in [25].
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Keywords: Galois group; $q$-difference equations; differential Tannakian categories; Kolchin differential groups.
@article{CML_2020__12_2_11_0, author = {Di Vizio, Lucia and Hardouin, Charlotte}, title = {Galois theories of $q$-difference equations: comparison theorems}, journal = {Confluentes Mathematici}, pages = {11--35}, publisher = {Institut Camille Jordan}, volume = {12}, number = {2}, year = {2020}, doi = {10.5802/cml.66}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/cml.66/} }
TY - JOUR AU - Di Vizio, Lucia AU - Hardouin, Charlotte TI - Galois theories of $q$-difference equations: comparison theorems JO - Confluentes Mathematici PY - 2020 SP - 11 EP - 35 VL - 12 IS - 2 PB - Institut Camille Jordan UR - http://archive.numdam.org/articles/10.5802/cml.66/ DO - 10.5802/cml.66 LA - en ID - CML_2020__12_2_11_0 ER -
%0 Journal Article %A Di Vizio, Lucia %A Hardouin, Charlotte %T Galois theories of $q$-difference equations: comparison theorems %J Confluentes Mathematici %D 2020 %P 11-35 %V 12 %N 2 %I Institut Camille Jordan %U http://archive.numdam.org/articles/10.5802/cml.66/ %R 10.5802/cml.66 %G en %F CML_2020__12_2_11_0
Di Vizio, Lucia; Hardouin, Charlotte. Galois theories of $q$-difference equations: comparison theorems. Confluentes Mathematici, Volume 12 (2020) no. 2, pp. 11-35. doi : 10.5802/cml.66. http://archive.numdam.org/articles/10.5802/cml.66/
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