## Invariance of recurrence sequences under a Galois group.(English)Zbl 0859.11014

Let $$F$$ be a Galois field of order $$q,k$$ a fixed positive integer and $$R= F^{k\times k} [D]$$ for $$D$$ an indeterminate. Let $$L$$ be a field extension of $$F$$ of degree $$k$$ with fixed basis $$B=\{\alpha, \alpha^q, \dots, \alpha^{q^{k-1}}\}$$ and identify $$L_F$$ with $$F^{k \times 1}$$. The $$F$$-vector space $$\Gamma_k(F)$$ of all sequences over $$F^{k \times 1}$$ is a left $$R$$-module. For any regular $$f(D)\in R$$, $$\Omega_k(f(D)) = \{S\in\Gamma_k(F) : f(D)S = 0\}$$ is a finite $$F[D]$$-module. The Galois group $$G(L/F)$$ is generated by $$\sigma(a) = a^q$$, $$a\in L$$. The question of the invariance of an $$\Omega_k(f(D))$$ under the Galois group is considered. A complete answer is given for the case $$k=2$$ and an explicit construction of a generating set and the dimension of an $$\Omega_2(f(D))$$ is given if $$f^\eta(D) = f(D)$$ for $$\eta$$ an inner automorphism of $$R$$. The case for $$k>3$$ remains unsolved.

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11T99 Finite fields and commutative rings (number-theoretic aspects) 15A24 Matrix equations and identities 16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
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