Hilbert’s tenth problem over the rational numbers
We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.
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this was arguably the biggest open problem in computability theory (note that uncomputability of solving Diophantine equations, i.e. polynomial equations over the integers , was proved in the 1970s, giving a negative answer to Hilbert’s 10th Problem) Any of the above, alone, could easily have been…
Recently, Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers was solved negatively by OpenAI based on a five-contact height estimate and a pole-parity construction. In this paper, we prove further that there is no algorithm to decide for any polynomial $P(x_1,\ldots,x_{86})$…
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A Brief Digest of Some New OpenAI Results | Persiflagegaloisrepresentations.orgPersiflage
As for Hilbert’s tenth problem itself, it seemed plausible to me that it was actually solvable over the rational numbers. I don’t think I would have put money on that, however. I still certainly hope that Hilbert’s tenth problem has a positive answer over the rational numbers for algebraic curves,…