Galois groups and connection matrices for \(q\)-difference equations. (English) Zbl 0844.12004

Summary: We study the Galois group of a matrix \(q\)-difference equation with rational coefficients which is regular at \(0\) and \(\infty\), in the sense of (difference) Picard-Vessiot theory, and show that it coincides with the algebraic group generated by matrices \(C(z) C(w)^{-1}\), \(z, w \in C^*\), where \(C(z)\) is the Birkhoff connection matrix of the equation.


12H10 Difference algebra
39A10 Additive difference equations
Full Text: DOI EuDML


[1] Richard M. Cohn, Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965. · Zbl 0127.26402
[2] P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111 – 195 (French). · Zbl 0727.14010
[3] Charles H. Franke, Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc. 108 (1963), 491 – 515. · Zbl 0116.02604
[4] Irving Kaplansky, An introduction to differential algebra, Actualités Sci. Ind., No. 1251 = Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. · Zbl 0083.03301
[5] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York-London, 1973. Pure and Applied Mathematics, Vol. 54. · Zbl 0264.12102
[6] Ritt, J.F., Differential algebra, Colloquium publ. of AMS, 1950. · Zbl 0037.18501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.