2020/12/17 by Albert Garreta, Garreta, Albert, Alexei Miasnikov +3
Mathematics · #11D72 #11D99 #11U05 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric and Algebraic Topology #Logic (math.LO) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2012.09787
openalex publication_date 2020/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study systems of polynomial equations in infinite finitely generated commutative associative rings with an identity element. For each such ring R we obtain an interpretation by systems of equations of a ring of integers O of a finite field extension of either ℚ or \mathbbFp(t), for some prime p and variable t. This implies that the Diophantine problem (decidability of systems of polynomial equations) in O is reducible to the same problem in R. If, in particular, R has positive characteristic or, more generally, if R has infinite rank, then we further obtain an interpretation by systems of equations of the ring \mathbbFp[t] in R. This implies that the Diophantine problem in R is undecidable in this case. In the remaining case where R has finite rank and zero characteristic, we see that O is a ring of algebraic integers, and then the long-standing conjecture that ℤ is always interpretable by systems of equations in a ring of algebraic integers carries over to R. If true, it implies that the Diophantine problem in R is also undecidable. Thus, in this case the Diophantine problem in every infinite finitely generated commutative unitary ring is undecidable. The present is the first in a series of papers were we study the Diophantine problem in different types of rings and algebras.