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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 $R$.)

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  • $\begingroup$ 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. $\endgroup$ Jan 26 '20 at 3:02
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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).

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  • $\begingroup$ Many thanks ! (Note: in your answer there are no polynomials x and y) $\endgroup$ Jan 26 '20 at 2:32

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