Premium
The Undecidability of the Unification Problem for Nilpotent Groups of Class ⩾ 5
Author(s) -
Burke E. K.
Publication year - 1993
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-48.1.52
Subject(s) - undecidable problem , nilpotent , unification , mathematics , nilpotent group , class (philosophy) , diophantine equation , central series , decidability , group (periodic table) , pure mathematics , algebra over a field , discrete mathematics , computer science , physics , quantum mechanics , artificial intelligence , programming language
We study the unification problem for the theories of nilpotent groups of class at least 5. This is equivalent to the problem of constructing an algorithm which will solve any equation in a free nilpotent group of class at least 5. Romankov in 1977 showed that the ‘endomorphic reducibility’ problem is undecidable for free nilpotent groups of class at least 9 and it follows that the Unification Problem for the theories of nilpotent groups of class at least 9 is undecidable. We shall improve this to 5 by reducing the problem of algorithmically solving an arbitrary diophantine equation of degree 4 to that of solving equations in a free nilpotent group of class 5 and then appealing to Matiyasevitch's Theorem.