What is the computational complexity of Hilbert Tenth problem over $\mathbb{Q_p}$

We know that Hilbert Tenth problem over $\mathbb{Q_p}$ is decidable,so what is the computational complexity of Hilbert Tenth problem over $\mathbb{Q_p}$?

Is it equivalent to Tarski elimination algorithm or lower?

• By Qp you mean positive rationals? Do you have a reference? I was under the impression that it's open for (general) rationals. – Mikolas Nov 12 '17 at 1:31
• @Mikolas Hilbert Tenth problem over $\mathbb{Q_p}$ is decidable, but Hilbert Tenth problem over $\mathbb{Q}$ is open. – XL _At_Here_There Nov 12 '17 at 2:54
• OK, do you have a reference for that? – Mikolas Nov 12 '17 at 12:40
• @Mikolas I assume XL_at_China means the p-adic rationals. – Joshua Grochow Nov 12 '17 at 19:14
• This seems to be the original reference, maybe someone can extract a running time estimate from it projecteuclid.org/euclid.bams/1183525364. – Sasho Nikolov Nov 18 '17 at 18:28