Generalizations of two theorems of Ritt on decompositions of polynomial maps

Abstract

Two theorems of J. F. Ritt on decompositions of polynomials maps are generalized to a more general situation: for, so-called, reduction monoids ($(K[x],\circ )$ and $(K[x^2]x,\circ )$ are examples of reduction monoids). In particular, analogues of the two theorems of J. F. Ritt hold for the monoid $(K[x^2]x,\circ )$ of odd polynomials. It is shown that, in general, the two theorems of J. F. Ritt fail for the cusp $(K+K[x]x^2,\circ )$ but their analogues are still true for decompositions of maximal length of regular elements of the cusp.

31 pages, the definition of the monoid O is corrected, reference is added