Reducing the polynomial-like iterative equations order and a generalized Zolt\'an Boros' problem

We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive iterates of the unknown function at a given point. As an application we solve a problem of Zolt'an Boros posed during the 50th ISFE (2012).