# Classification of associative polynomial functions

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 $$R$$.)

• If you find an associative polynomial P(x,y) where the term Q(x,y) = P(x,P(y,x)) is also associative, then the structure with underlying set R and binary operation P generates a locally finite variety; further, you just have to check three more terms in P to see if they and all other terms in P are associative. In case you needed lots of associative polynomials. Gerhard "Search On Hyperassociativity In MathOverflow" Paseman, 2020.01.25. Jan 26 '20 at 3:02

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".
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).