Archiv der Mathematik

, Volume 97, Issue 2, pp 115–124

# A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

• Erhard Aichinger
• Stefan Steinerberger
Article

## Abstract

Fried and MacRae (Math. Ann. 180, 220–226 (1969)) proved that for univariate polynomials $${p,q, f, g \in \mathbb{K}[t]}$$ ($${\mathbb{K}}$$ a field) with p, q nonconstant, p(x) − q(y) divides f(x) − g(y) in $${\mathbb{K}[x,y]}$$ if and only if there is $${h \in \mathbb{K}[t]}$$ such that f = h(p(t)) and g = h(q(t)). Schicho (Arch. Math. 65, 239–243 (1995)) proved this theorem from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In the present note, we give a new proof of one of these generalizations. The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions f, g from $${\mathbb{C}}$$ to $${\mathbb{C}}$$ : if both the composition $${f \circ g}$$ and g are polynomial functions, then f has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. This provides a way to prove a generalization of a result by Carlitz (Acta Sci. Math. (Szeged) 24, 196–203 (1963)) that describes those univariate polynomials over finite fields that induce bijective functions on all of their finite extensions.

13B25 (12E05)

## Keywords

Polynomial composition Multivariate Fried-MacRae-theorems Injective polynomials

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