What is known about a classification of associative (binary) polynomial functions? First of all, it is interesting in two cases: over Integral domain (or even over field) and over ring of integers modulo $n$.


(A polynomial binary function is function $R \times R \to R$ induced by a polynomial in two variables $P$ over a ring $R$.)



Classification of associative polynomial functions
Over an infinite integral domain you can classify all polynomials that satisfy the associativity functional equation. A quick degree consideration of both sides tells you that the polynomial is at most degree $1$ in each variable, so the only answers are $x,y, c+x+y$ and $c\_1(x+c\_2)(y+c\_2)-c\_2$. In fact you can classify all $n$-variable polynomials which satisfy the $n$-ary version of associativity, as done in the paper ["A description of n-ary semigroups polynomial-derived from integral domains"](https://arxiv.org/abs/1011.6291v2).


As far as finite rings, I doubt you can say anything meaningful. For example, already over $\mathbb Z/p\mathbb Z$, any binary function can be written as a polynomial map, so you would be asking for a classification of all associative operations on this set (this number increases [very fast](https://oeis.org/A023814)).


