2018/05/07 by Albert Garreta, Garreta, Albert, Alexei Miasnikov +3
Computer Science · Mathematics · #03B25 #03C60 #03D35 #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Polynomial and algebraic computation #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1805.02573
openalex publication_date 2018/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring R (or algebra) in one of these classes 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 Karp-reducible to the same problem in R. In several cases we further obtain an interpretation by systems of equations of the ring \mathbbFp[t] in R, which implies that the Diophantine problem in R is undecidable in this case. Otherwise, the ring O is a ring of algebraic integers, and then the long-standing conjecture that ℤ is always interpretable by systems of equations in O carries over to R. If true, it implies that the Diophantine problem in R is also undecidable. Some of the classes of f.g. rings studied in this paper are the following: all associative, commutative, non-unitary rings (a similar statement for the unitary case was obtained by Eisentraeger); all possibly non-associative, non-commutative non-unitary rings that are f.g. as an abelian group; and several classes of f.g. non-commutative rings. Analogous statements are obtained for algebras over f.g. associative commutative unitary rings. Another contribution is the technique by which the aforementioned results are obtained: We show that given a bilinear map f: A× B → C between f.g. abelian groups (or modules), under mild assumptions, there exists a certain ring (or algebra) R with nice properties which is interpretable by systems of equations in the multi-sorted structure (A,B,C;f). This result is not only relevant for rings and algebras, but also in other structures such as groups, as demonstrated previously by the authors.