## Commuting Polynomials |

**Karl Zimmermann**

Union College

May 30, 2001

3:30 PM

Bailey Hall 106

It is well know that polynomials with real number coefficients commute under addtion [] and under multiplication [ f(x) +g(x) =g(x) +f(x)]. Composition of polynomials is a different matter, and more often than not, f(x)g(x) =g(x)f(x)g(f(x)) does not equalf(g(x)). Nonetheless,and f(x) = 2x+x^{2}do commute under composition and you can check to see that g(x) = 3x+ 2x^{2}+x^{3}g(f(x)) = 6x+ 15x^{2}+ 20x^{3}+ 15x^{4}+ 6x^{5}+x^{6}=f(g(x)). It's a long computation by hand, but very fast using`Mathematica`

. (Try to construct other examples.)In this talk, we consider not just examples of two isolated polynomials that commute under composition, but examples of infinite sets of polynomials in which each pair commutes.

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