2021/03/09 by Zhi‐Wei Sun, Sun, Zhi-Wei
Mathematics · #03D25 #03D35 #11D99 #11U05 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2103.08302
openalex publication_date 2021/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we first review the history of Hilbert's Tenth Problem, and then study mixed quantifier prefixes over Diophantine equations with integer variables. For example, we prove that ∀2∃4 over \mathbb Z is undecidable, that is, there is no algorithm to determine for any P(x1,…,x6)∈\mathbb Z[x1,…,x6] whether ∀ x1∀ x2∃ x3∃ x4∃ x5∃ x6(P(x1,…,x6)=0), where x1,…,x6 are integer variables. We also have some similar undecidable results with universal quantifies bounded, for example, ∃2∀2∃2 over \mathbb Z with ∀ bounded is undecidable. We conjecture that ∀2∃2 over \mathbb Z is undecidable.