Matiyasevich's single-fold conjecture: every r.e. set has a single-fold Diophantine representation

Theorem 1.1: every r.e. $S\subseteq\mathbb N^n$, $n\ge1$, has a polynomial $P\in\mathbb Z[A_1..A_n,W_1..W_m]$ with $\#\{w\in\mathbb N^m:P(a,w)=0\}$ equal to 1 for $a\in S$ and 0 otherwise. Hence the finite-fold conjecture holds, and Diophantine solvability over $\mathbb N$ stays undecidable under an at-most-one-solution promise (Section 8). The construction bounds a Pell-type exponential with a cutoff from rational points on a rank-one elliptic curve. Witnesses range over $\mathbb N$; the paper does not claim the analogue over $\mathbb Z$ beyond what follows from it, nor any bound on degree or number of variables.

Theorem 1.1: every r.e. $S\subseteq\mathbb N^n$, $n\ge1$, has a polynomial $P\in\mathbb Z[A_1..A_n,W_1..W_m]$ with $\#\{w\in\mathbb N^m:P(a,w)=0\}$ equal to 1 for $a\in S$ and 0 otherwise. Hence the finite-fold conjecture holds, and Diophantine solvability over $\mathbb N$ stays undecidable under an at-most-one-solution promise (Section 8). The construction bounds a Pell-type exponential with a cutoff from rational points on a rank-one elliptic curve. Witnesses range over $\mathbb N$; the paper does not claim the analogue over $\mathbb Z$ beyond what follows from it, nor any bound on degree or number of variables.