Iterative \(q\)-difference Galois theory.

*(English)*Zbl 1203.12004In 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.

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.

Reviewer: Jacques Sauloy (Toulouse)

##### MSC:

12H10 | Difference algebra |

12H05 | Differential algebra |

39A13 | Difference equations, scaling (\(q\)-differences) |

##### Keywords:

Difference Galois theory; q-difference equations; Picard-Vessiot theory; higher derivations; Grothendieck-Katz conjecture##### References:

[1] | Ann. Sci. E’c. Norm. Sup. 34 (4) pp 685– (2001) |

[2] | André Y., Astérisque 296 pp 55– (2004) |

[3] | DOI: 10.2307/2373897 · Zbl 0374.12014 · doi:10.2307/2373897 |

[4] | Chatzidakis Z., London Math. Soc. Lect. Note Ser. 349 pp 73– (2008) |

[5] | Deligne P., Progr. Math. 87 pp 111– (1990) |

[6] | DOI: 10.1007/s00222-002-0241-z · Zbl 1023.12004 · doi:10.1007/s00222-002-0241-z |

[7] | Hasse H., Math. 177 pp 215– (1937) |

[8] | Marotte F., Ann. Inst. Fourier 50 (6) pp 1859– (2000) |

[9] | B., Preprint IWR pp 2001– (2001) |

[10] | Matzat B. H., Math. 557 pp 1– (2003) |

[11] | Sauloy J., Ann. Sci. E’c. Norm. Sup. 36 (4) pp 925– (2004) |

[12] | Astérisque 296 pp 191– (2004) |

[13] | Compos. Math. 97 (1) pp 227– (1995) |

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.