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Equation‐Solving in Free Nilpotent groups Class 2 and 3
Author(s) -
Truss J. K.
Publication year - 1995
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/27.1.39
Subject(s) - undecidable problem , decidability , mathematics , diophantine equation , unification , nilpotent , class (philosophy) , pure mathematics , integer (computer science) , quadratic equation , discrete mathematics , conjecture , nilpotent group , algebra over a field , computer science , geometry , artificial intelligence , programming language
This paper presents an improvement to the results of [1, 7, 8], where it was shown that the unification problem for nilpotent groups of class k is undecidable, for k ⩾ 9 in [8], and for k ⩾ 5 in [1, 7]. The value given is k ⩾ 3. The method of proof is a straightforward modification of those used previously, obtained by observing that it is an immediate consequence of Matiyasevich's Theorem that the problem of determining the solubility of a general system of quadratic diophantine equations is undecidable. We conjecture that the result is best possible, in the sense that it is believed that the unification problem for nilpotent groups of class 2 is decidable. It was shown in [2] how to reduce the class 2 unification problem to the solution of certain special systems of quadratic diophantine equations, and we show here that this is a decidable question for the case of two unknowns. A final solution will depend on exactly how these systems of equations behave in general.