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Results. We have constructed some rather general algorithms, which can (in theory) be applied in diverse situations. Let us describe them. A. ARITHMETIC GROUPS. Let G be an algebraic subgroup of GLn, defined by an explicitly given finite set of polynomial equations over Q in the matrix entries; suppose T is an explicitly given arithmetic subgroup in G: by this we mean (i) T is a subgroup of finite index in G z , (ii) an upper bound for this index is given, and (iii) we have an effective procedure to decide for each g E Gz whether or not g E.T. Algorithm Al finds finitely many matrices yx, . . . , ys G r and a finite set R of relations satisfied by them such that r = (yt, . . . , ys \ R > is a presentation of T. Suppose now that we are also given rational functions over Q, in the matrix entries, which define a rational representation p : G —> GLm, for some m. Algorithm A2 determines for any two vectors a and b in Q whether there exists y E F such that ap(y) = b; if so, it also finds such a matrix 7. The algorithms proceed by constructing "fundamental sets" for Gz in GR (and for Hz in HR, where H is the stabilizer of a in G in Algorithm A2), after the manner of Borel and Harish-Chandra, [BH] and [Bo]. Of course these constructions have to be made effective, which involves a certain amount of technical difficulty. There is then a fairly standard way to complete Al. The basic idea for A2, somewhat oversimplified, is as follows. One first decides whether there exists x E GR with ap(x) = 6, and if so, computes a sequence of rational approximations to such a matrix x\ this is possible by Tar ski's decision method for elementary algebra over R [T]. Then one constructs a "fundamental set" Q, with the following properties: (i) SIHZ = HR, (ii) £2 C Scx U • • • U USck

The solubility of certain decision problems in arithmetic and algebra