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Algebraic groups as difference Galois groups of linear differential equations. (English) Zbl 1475.12011

Let \(F\) denote a \(\delta\sigma\)-field of characteristic zero, i.e. a field with two commuting operators, a derivation \(\delta\) and an endomorphism \(\sigma\). For such fields it is possible to build the theory of a similar Kolchin’s Differential Galois theory. The authors in a number of publications [A. Bachmayr et al., Doc. Math. 23, 241–291 (2018; Zbl 1436.12006); Adv. Math. 381, Article ID 107605, 28 p. (2021; Zbl 1461.12003); Trans. Am. Math. Soc. 374, No. 6, 4293–4308 (2021; Zbl 1481.12008)] make contribution to the development of such theory. This paper discusses the converse problem of the \(\sigma\)-Picard-Vessiot theory. In the usual Picard-Vessiot theory, for the field \(\mathbb{C}(x)\) and the linear algebraic group, the converse problem always has a solution [C. Tretkoff and M. Tretkoff, Am. J. Math. 101, 1327–1332 (1979; Zbl 0423.12021)]. The authors show that not every difference algebraic group occurs as a \(\sigma\)-Galois group of a \(\sigma\)-Picard-Vessiot extension of \(\mathbb{C}(x)\) . But the main result of the paper says that there are still quite a lot of positive occasions.
Theorem. Every linear algebraic group over \(\mathbb{C}\), considered as a difference algebraic group, occurs as a \(\sigma\)-Galois group over \(\mathbb{C}(x)\).

MSC:

12H10 Difference algebra
12H05 Differential algebra
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
14L15 Group schemes
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References:

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