(Both the published version and the arXiv version are available online.)

Let g and h be nonconstant polynomials over a field K. We say that g and h have a common composite if there are nonconstant polynomials u and v over K such that u(g(x)) = v(h(x)). If K has characteristic zero, one can describe all pairs of polynomials having a common composite, as a consequence of classical results due to Ritt. We study the analogous problem in positive characteristic.

We prove that, if two polynomials have a common composite, then the minimal degree of any common composite is a power of the characteristic times the least common multiple of the degrees of the polynomials. Moreover, the existence of a common composite over an extension of K implies the existence of a common composite over K.

Our main results give necessary and sufficient criteria for the existence of a common composite, as well as various necessary conditions which enable us to exhibit several pairs of polynomials having no common composite.

Michael Zieve:  home page   publication list