Positive existential definability of multiplication from addition and the range of a polynomial

We consider the problem of recovering multiplication in the integers from enrichments of its additive structure, in the positive existential context. We prove that if a conjecture by Caporaso–Harris–Mazur holds, then for all integer-valued polynomials F of degree at least 2, multiplication is positive-existentially definable in (Z; 0, 1,+, R F , =) where R F is the unary relation F (Z). Similar results were only known for the polynomials F ( t ) = t 2 (under the Bombieri–Lang conjecture) and F ( t ) = t n (under a generalization of the abc conjecture).