On linear polynomials commutable with a polynomial. (Über die mit einem Polynom vertauschbaren linearen Polynome.) (German) Zbl 0741.12003

Eigenthaler, Günther; Nöbauer, Wilfried

Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 199, No. 4-7, 143-153 (1990).

Let \(K[x]\) be the polynomial ring over a field \(K\), and \(\langle K[x],\circ\rangle\) the semigroup under composition. The authors investigate the so-called linear centralizer \(LZ(f(x))=L\cap Z(f(x))\), \(L\) denoting the group of units of \(K[x]\), \(L=\{ax+b,a,b,\in K,a\neq 0\}\) and \(Z(f(x))\) being the centralizer of \(f(x)\), \(Z(f(x))=\{g(x)\in K[x]\backslash K\), \(f(x)\circ g(x)=g(x)\circ f(x)\}\). It is shown that there exists an \(\ell(x)\in L\) such that for \(f_ 1(x)=\ell^{- 1}(x)\circ f(x)\circ\ell(x)\) \(LZ(f_ 1(x))=T\circ S\), where \(T,S\) are subgroups of \(\{x+a,a\in K\}\) and \(\{bx,b\in K^*\}\), respectively. Furthermore it is proved that mostly \(T\) is trivial, in particular if \(\text{char }K=0\). The case \(\text{char }K=p\) is dealt with, too.

Reviewer: G.Kowol (Wien)

MSC:

12E05	Polynomials in general fields (irreducibility, etc.)
12E10	Special polynomials in general fields
13B25	Polynomials over commutative rings
20E99	Structure and classification of infinite or finite groups

Keywords:

composition of polynomials; commutable polynomials; linear centralizer