Submission date: 6 April 2021

Abstract:

In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by proving that ℚ ∖ ℤ is diophantine over ℚ, i.e., there is a polynomial P(t,x_1,...,x_n) in ℚ[t,x_1,\ldots,x_n] such that for any rational number t we have t ∉ ℤ ⇔ ∃ x_1 ... ∃ x_n[P(t,x_1,...,x_n)=0] where variables range over ℚ, equivalently t in ℤ ⇔ ∀ x_1 ... ∀ x_n[P(t,x_1,...,x_n) ≠ 0]. In this paper we prove further that we may even take n=32 and require deg P<6 × 10^{11}, which provides the best record in this direction. Combining this with a result of Sun, we get that there is no algorithm to decide for any P(x_1,...,x_{41}) in ℤ[x_1,...,x_{41}] whether ∀ x_1 ... ∀ x_9 ∃ y_1 ... ∃ y_{32}[P(x_1,...,x_9,y_1,...,y_{32})=0].

Mathematics Subject Classification: 03D35, 11U05, 03D25, 11D99, 11S99

Keywords and phrases:

Full text arXiv 2104.02520: pdf, ps.