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Iterative \(q\)-difference Galois theory. (English) Zbl 1203.12004
In positive characteristic, the usual Taylor formula: \(f(x+y) = \sum \frac{f^{(k)}(x)}{k!} y^k\) fails because of the vanishing denominators \(k !\). Algebraic geometers (like Miyanishi) have overcome this difficulty by replacing the divided iterates of the derivation operator by so-called higher derivations, a family of operators that satisfy axiomatically the same laws. Using these higher derivations, Matzat and van der Put succeeded in building a differential Galois theory for positive characteristic.
In the case of \(q\)-difference equations over the complex numbers, similar difficulties arise when \(q\) is a root of unity. To take an example, the binomial formula for \(q\)-commuting variables \(x,y\) such that \(y x = q x y\) reads: \[ (x+y)^n = \sum_{k=0}^n {n \choose k}_q x^{n-k} y^k, \] where we successively define \(q\)-integers, \(q\)-factorials and \(q\)-binomial coefficients: \[ \left[n\right]_q := \dfrac{q^n - 1}{q-1}, \quad \left[n\right]!_q := \prod_{k=1}^n \left[n\right]_q, \quad {n \choose k}_q := \dfrac{\left[n\right]!_q}{\left[k\right]!_q \left[n-k\right]!_q} \cdot \] The \(q\)-binomial theorem is a \(q\)-analog of (an instance of) Taylor formula in the following way. One defines the \(q\)-derivation operator by the formula: \[ \delta_q f(x) := \dfrac{f(q x) - f(x)}{(q - 1) x} \cdot \] Then one finds: \[ {n \choose k}_q x^{n-k} = \dfrac{\delta_q^k(x^n)}{\left[k\right]!_q } \cdot \] Of course, all of this fails to make sense if \(q\) is a root of unity, because of the vanishing denominators \(\left[k\right]!_q\).
In the paper under review, the author defines axiomatically what she calls iterative \(q\)-difference operators over a \(\mathbb{C}(t)\)-algebra (the use of difference rings instead of fields is a classical necessity in difference Galois theory). She builds on that basis a coherent and flexible Galois theory and uses it in concrete substantial examples, including a new proof of some results by Hendricks. Last, she links her theory to the \(q\)-analog of the Grothendieck-Katz conjecture previously studied by Di Vizio.
The author asked the present reviewer to point out an omission in definition 2.4: the restriction of each \(\delta_R^{(k)}\) to \(\mathbb{C}(t)\) should coincide with the operator \(\delta_q^{(k)}\) as it appears in definition 2.8 The reviewer wishes to point out another minor mistake: in definition 2.1, the \(q\)-binomial operator cannot be directly defined by a division when \(q\) is a root of unity. One should first take for \(q\) an indeterminate, then show that the definition actually yields a polynomial, last specialize \(q\) to a numerical value.

12H10 Difference algebra
12H05 Differential algebra
39A13 Difference equations, scaling (\(q\)-differences)
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