##
**Galois theory of difference equations.**
*(English)*
Zbl 0930.12006

Lecture Notes in Mathematics. 1666. Berlin: Springer. vii, 180 p. (1997).

A difference field is a field \(k\) with an automorphism \(\phi\). A difference operator of order \(n\) over \(k\) is an expression of the form \(L(Y)=\phi^n(Y)+a_{n-1}\phi^{n-1}(Y)+ \dots+a_0Y\), \(a_i \in k\). A difference algebra over \(k\) is a commutative \(k\) algebra \(R\) with a derivation \(\phi_R\) extending \(\phi\). (Generally, \(\phi_R\) is denoted \(\phi\).) A solution \(y \in R\) of the difference equation \(L=0\) is an element \(y\) such that \(L(y)=0\). In analogy with the corresponding theory for linear differential equations, the authors develop a Galois theory for these difference equations.

Recall that the analogue in differential Galois theory of the splitting fields of polynomial Galois theory are the Picard–Vessiot extensions. The authors show by elementary example that there can be no such field extensions in the difference case, and they define instead the Picard–Vessiot ring associated to the equation \(L=0\) above. This is a difference algebra \(R\) which contains a fundamental set of solutions for \(L=0\), has no proper difference ideals, and is minimal with respect to these properties. “Contains a fundamental set of solutions” means the following: as with differential equations, the order \(n\) equation \(L=0\) can be written as a matrix equation \(\phi(X)=AX\) where \(X\) is the \(n\) vector with \(i\)th entry \(\phi^i(y)\) and \(A\in\text{GL}_n(k)\). \(R\) contains a fundamental set of solutions if there is a matrix \(U\in\text{GL}_n(R)\) such that \(\phi(U)=AU\). More generally, the authors consider arbitrary matrix equations over \(k\) (any \(A\in\text{GL}_n(k))\); a fundamental matrix in \(R\) is a \(U\in\text{GL}_n(R)\) such that \(\phi(U)=AU\). A Picard–Vessiot ring for this is then as above, with the requirement that it contains a fundamental matrix.

The field of constants of \(k\) is \(C =\{a\in k\mid \phi(a)=a \}\). Under the assumption that \(C\) is algebraically closed, the authors prove that Picard–Vessiot rings exist and are unique. They show that the group \(G\) of difference automorphisms of a Picard–Vessiot ring \(R\) over \(k\) is an algebraic group over \(C\), and that \(R\) is the coordinate ring of a principal homogeneous space for \(G\). This gives a Galois correspondence for difference equations. They provide algorithms for the calculation of \(G\) in low order (\(n=1,2\)) cases, and they show that every connected linear algebraic group \(C\) occurs as a difference Galois group over \(C(z)\) (where \(\phi(z)=z+1\), and in addition \(C\) has characteristic zero). They apply their results to linear recursive and differentially finite sequences. They also analyze the special cases where \(k=\overline{\mathbf{F}}_p(x)\) and where \(k\) is the field of Puiseux series.

The authors also investigate the analytic case, where \(k\) is a field of convergent complex power series, and the solutions to the difference equation are to be meromorphic functions. Here the algebraic theory is regarded as giving symbolic solutions, and meromorphic function solutions are termed asymptotic. For a class of equations the authors call semi-regular, the asymptotic lifts are generally unique, and the difference Galois group determined. For the class the authors call mild, there are unique asymptotic lifts subject to some additional conditions, and again the difference Galois group can be determined. And the authors deal with the remaining class of equations, the wild equations, as well.

They also include a chapter on \(q\)-difference equations. Here \(q\) is a non-zero non-root-of-unity complex number with logarithm \(\ell=2\pi i \tau\), \(k\) denotes the union of the fields \({\mathbf{C}}(z^{1/m})\) with automorphism \(\phi(z^\lambda)=e^{\ell \lambda}z^\lambda\), and the corresponding difference equations are over \(k\).

The book has a bibliography of 63 items and an index.

Recall that the analogue in differential Galois theory of the splitting fields of polynomial Galois theory are the Picard–Vessiot extensions. The authors show by elementary example that there can be no such field extensions in the difference case, and they define instead the Picard–Vessiot ring associated to the equation \(L=0\) above. This is a difference algebra \(R\) which contains a fundamental set of solutions for \(L=0\), has no proper difference ideals, and is minimal with respect to these properties. “Contains a fundamental set of solutions” means the following: as with differential equations, the order \(n\) equation \(L=0\) can be written as a matrix equation \(\phi(X)=AX\) where \(X\) is the \(n\) vector with \(i\)th entry \(\phi^i(y)\) and \(A\in\text{GL}_n(k)\). \(R\) contains a fundamental set of solutions if there is a matrix \(U\in\text{GL}_n(R)\) such that \(\phi(U)=AU\). More generally, the authors consider arbitrary matrix equations over \(k\) (any \(A\in\text{GL}_n(k))\); a fundamental matrix in \(R\) is a \(U\in\text{GL}_n(R)\) such that \(\phi(U)=AU\). A Picard–Vessiot ring for this is then as above, with the requirement that it contains a fundamental matrix.

The field of constants of \(k\) is \(C =\{a\in k\mid \phi(a)=a \}\). Under the assumption that \(C\) is algebraically closed, the authors prove that Picard–Vessiot rings exist and are unique. They show that the group \(G\) of difference automorphisms of a Picard–Vessiot ring \(R\) over \(k\) is an algebraic group over \(C\), and that \(R\) is the coordinate ring of a principal homogeneous space for \(G\). This gives a Galois correspondence for difference equations. They provide algorithms for the calculation of \(G\) in low order (\(n=1,2\)) cases, and they show that every connected linear algebraic group \(C\) occurs as a difference Galois group over \(C(z)\) (where \(\phi(z)=z+1\), and in addition \(C\) has characteristic zero). They apply their results to linear recursive and differentially finite sequences. They also analyze the special cases where \(k=\overline{\mathbf{F}}_p(x)\) and where \(k\) is the field of Puiseux series.

The authors also investigate the analytic case, where \(k\) is a field of convergent complex power series, and the solutions to the difference equation are to be meromorphic functions. Here the algebraic theory is regarded as giving symbolic solutions, and meromorphic function solutions are termed asymptotic. For a class of equations the authors call semi-regular, the asymptotic lifts are generally unique, and the difference Galois group determined. For the class the authors call mild, there are unique asymptotic lifts subject to some additional conditions, and again the difference Galois group can be determined. And the authors deal with the remaining class of equations, the wild equations, as well.

They also include a chapter on \(q\)-difference equations. Here \(q\) is a non-zero non-root-of-unity complex number with logarithm \(\ell=2\pi i \tau\), \(k\) denotes the union of the fields \({\mathbf{C}}(z^{1/m})\) with automorphism \(\phi(z^\lambda)=e^{\ell \lambda}z^\lambda\), and the corresponding difference equations are over \(k\).

The book has a bibliography of 63 items and an index.

Reviewer: Andy R. Magid (Norman)

### MSC:

12H10 | Difference algebra |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

39A10 | Additive difference equations |