Undecidability of equations in free Lie algebras

Kharlampovich, Olga; Myasnikov, Alexei

doi:10.1090/tran/7579

Cited by 5 publications

(6 citation statements)

References 30 publications

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“…Then the conclusions of Theorem 1.6 hold for F . This complements the aforementioned results of Romankov [37] and of Kharlampovich and Miasnikov [15,17] regarding free algebras. We remark that in [37] it is proved (among others) that the algebras of Corollary 5.10 actually have undecidable Diophantine problem if Λ = Z.…”

Section: Introductionsupporting

confidence: 89%

“…Corollary 5.10 complements Romankov's [37], and Kharlampovich and Miasnikov's [15,17] papers on free algebras. In the first reference it is proved that D(F ; L ring ) is undecidable for many types of free rings F .…”

Section: Subrings Of Global Fieldssupporting

confidence: 67%

“…Hence each element as in (17) If R is associative, then any element x ∈ I n is also a sum of elements of the form (17), and the proof follows as in the non-associative case.…”

Section: Finitely Generated Rings and Algebras Satisfying An Infinite...mentioning

confidence: 93%

“…Regarding non-commutative rings, Romankov [37] showed that D(F ) is undecidable in several types of free rings F , which include free Lie rings, free associative or nonassociative rings, and free nilpotent rings. In the same direction the second author, in collaboration with Kharlampovich, recently proved undecidability of D(A) in the language of rings, for A any of the following: a free associative k-algebra, a free Lie k-algebra (of rank at least 3), and many group k-algebras [15,17]. In all these cases k is an arbitrary field.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers

Garreta¹,

Miasnikov²,

Ovchinnikov³

2018

Preprint

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We study systems of equations in different families of rings and algebras. In each such structure R we interpret by systems of equations (e-interpret) a ring of integers O of a global field. The long standing conjecture that Z is always e-interpretable in O then carries over to R, and if true it implies that the Diophantine problem in R is undecidable. The conjecture is known to be true if O has positive characteristic, i.e. if O is not a ring of algebraic integers. As a corollary we describe families of structures where the Diophantine problem is undecidable, and in other cases we conjecture that it is so. In passing we obtain that the first order theory with constants of all the aforementioned structures R is undecidable.

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Section: Introductionsupporting

confidence: 89%

Section: Subrings Of Global Fieldssupporting

confidence: 67%

“…Hence each element as in (17) If R is associative, then any element x ∈ I n is also a sum of elements of the form (17), and the proof follows as in the non-associative case.…”

Section: Finitely Generated Rings and Algebras Satisfying An Infinite...mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers

Garreta¹,

Miasnikov²,

Ovchinnikov³

2018

Preprint

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“…One can view these rings as free Z-algebras, it is essentail, since the proofs use undecidability of the Diophantine problem in the coefficients Z. Using different methods Kharlampovich and Miasnikov recently proved undecidability of DpAq, for any of the following rings A: a free associative k-algebra, a free Lie k-algebra (of rank at least 3), and group k-algebras kpGq for various groups G (including free, torsion-free hyperbolic, right-angled Artin, and other groups) [19,20]. In all these results the field k is arbitrary, possibly with decidable Dpkq.…”

Section: Introductionmentioning

confidence: 99%

The Diophantine problem in finitely generated commutative rings

Garreta,

Miasnikov,

Ovchinnikov

2020

Preprint

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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 Fpptq, 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 Fprts 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.

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