Hilbert's tenth problem over the rational numbers

Claims a negative answer: rational solvability of integer polynomial equations (variables unbounded) is undecidable, of Turing degree $0'$, and stays so for total degree at most four. The method reduces integer solvability to finitely many rational-solvability tests per input, using rank-one elliptic curves, a five-point height estimate and a parity condition from 2-descent. It does not define $\mathbb Z$ existentially in $\mathbb Q$ (consistent with Mazur's conjecture), does not bound the number of variables, and does not address other number fields beyond what is known.

Claims a negative answer: rational solvability of integer polynomial equations (variables unbounded) is undecidable, of Turing degree $0'$, and stays so for total degree at most four. The method reduces integer solvability to finitely many rational-solvability tests per input, using rank-one elliptic curves, a five-point height estimate and a parity condition from 2-descent. It does not define $\mathbb Z$ existentially in $\mathbb Q$ (consistent with Mazur's conjecture), does not bound the number of variables, and does not address other number fields beyond what is known.