Abstract
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,+, RF, =) where RF is the unary relation F(Z). Similar results were only known for the polynomials F(t) = t2 (under the Bombieri–Lang conjecture) and F(t) = tn (under a generalization of the abc conjecture).
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