Q∖Z is diophantine over Q with 32 unknowns

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,x1,…,xn)∈Z[t,x1,…,xn] such that for any rational number t we have t/∈Z⟺∃x1⋯∃xn [P(t,x1,…,xn)=0], where variables range over Q, equivalently t∈Z⟺∀x1⋯∀xn [P(t,x1,…,xn)/=0]. In this paper we prove that we may take n=32. Combining this with a result of Z.-W. Sun, we show that there is no algorithm to decide for any f(x1,…,x41)∈Z[x1,…,x41] whether ∀x1⋯∀x9∃y1⋯∃y32 [f(x1,…,x9,y1,…,y32)=0], where variables range over Q.

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).