Rigidity conditions for binomials

Source abstract

We consider the following three conditions for polynomials in several variables. Condition \textbf{(1)} holds for a polynomial if, for every affine integral domain, the following is true: whenever the result of substituting algebraically independent elements into the polynomial lies in the kernel of a locally nilpotent derivation, the elements themselves must lie in that kernel. A polynomial satisfies condition \textbf{(2)} if it does not belong to the kernel of any nonzero LND of the polynomial algebra. It satisfies condition \textbf{(3)} if the quotient algebra modulo the ideal generated by the polynomial is rigid, that is, it admits no nontrivial LNDs. For irreducible binomials, we prove that all three conditions are equivalent. For reducible binomials, we establish all existing implications between these conditions.