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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.

##### MSC:
 12H10 Difference algebra 12H05 Differential algebra 39A13 Difference equations, scaling ($$q$$-differences)
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