Hilbert’s tenth problem over the rational numbers
Source: github.com/openai/math/tree/main/preprints/Hilberts-tenth-p…
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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5) Hilbert's Tenth Problem over \(\mathbb{Q}\) Hilbert's Tenth Problem over \(\mathbb{Z}\), the standard version, is the following decision problem: Find an algorithm that will, given a polynomial \(p\in \mathbb{Z}[x_1,\ldots,x_n]\), determine whether there are integers \(a_1,\ldots,a_n\in…
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…
There are 31 "result families" in the Number Theory section of the OpenAI research catalog. Of these, I think it's likely that all of them are with conventional wisdom, in the sense that >50% of experts thought that the statement was >50% likely to be true. In other words, there are no examples of…
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.orgFrank Calegari
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,…