R. Beals. Algorithms for matrix groups and the Tits alternative. In Proceedings of 36th Annual Symposium on Foundations of Computer Science, 1995, pp. 593-602.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... char K n then a can be computed in a obvious way. In the remaining cases rst we compute the squarefree part of the characteristic polynomial of a and use the Newton Hensel iteration for computing a s . The detailed description as well as an analysis of the algorithm is given in [BBCIL] or [Be]. It is shown that the method requires n O(1) arithmetical operations, and the Boolean cost is also a polynomial of the input size provided that systems of linear equations over K can be solved in polynomial time. By a torus over K or a K torus we mean a nite dimensional commutative algebra ....

R. Beals, Algorithms for matrix groups and the Tits alternative, Proc. 36th IEEE FOCS, (1995), 593-602.

....for G. In [Ge] it is proved that the run time of his algorithm is polynomial in the length of the input. In the case when G GL(n; Z) and Env(G) is a field, this author independently discovered an algorithm to find a presentation for G. See Section 2 for a discussion of these two algorithms. In [Be1] Beals describes an algorithm for deciding whether or not a given finitely generated subgroup of GL(n; R) is solvable by finite. He proves that the run time of his algorithm is polynomial in the length of input. His algorithm appears to be suitable for computer implementation. We describe in ....

....computer implementation. Further experimentation is needed to determine the range of input for which they are practical. In the case of the algorithm to decide whether or not a given matrix group is solvableby finite, experiments are needed to compare the efficiency of our algorithm with that in [Be1]. Acknowledgements. This work was undertaken as part of the author s Ph. D. thesis under the direction of Charles Sims. The author would also like to thank Jerrold Tunnell for help with the algebraic number theory as well as Eddie Lo for many stimulating conversations about polycyclic groups. ....

R. Beals, Algorithms for matrix groups and the Tits alternative, Proc. 36th IEEE FOCS (1995), 593-602.

....or not G is polycyclic by finite. For polycyclic by finite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We also describe an algorithm for deciding whether or not G is solvable by finite, providing an alternative to the algorithm proposed by Beals ([Be1]) for this problem. Baumslag, Cannonito, Robinson and Segal prove that the problem of determining whether or not a finitely generated subgroup of GL(n; Z) is polycyclic by finite is decidable and that the problem of testing membership in a polycyclic by finite subgroup of GL(n; Z) is also ....

....the input. They also describe an alternative method for testing membership in a matrix group which is both unipotent and abelian. Both the algorithms described here and those in [BBCIL] rely heavily on the work of Ge ( Ge] concerning algorithms for multiplicative subgroups of a number field. In [Be1] Beals describes an algorithm for deciding whether or not a given finitely generated subgroup of GL(n; R) is solvable by finite. He proves that the run time of his algorithm is polynomial in the length of input. His algorithm appears to be suitable for computer implementation. We describe here an ....

Beals, Robert (1995). Algorithms for matrix groups and the Tits alternative. To appear, Proc. 36th IEEE FOCS.

....if ag Gamma1 2 K. This approach might be particularly efficient in the case when G is polycyclic, since then we can use algorithms developed for working with solvable matrix groups over finite fields. See [Lu] The second algorithm for deciding whether or not a 2 G relies on the algorithms in [Be1] for working with abelian by finite matrix groups as well as the algorithm in Section 2.3 for testing membership in a unitriangular matrix group. Recall that K G, jG : Kj 1, and K is triangularizable. Find a basis for R n relative to which K has the block structure described in Proposition ....

.... Delta Delta Delta 0 Delta Delta Delta Delta 0 0 0 Delta Delta 0 a r 1 C C C C C C C C C C C C C C C A We can obtain generators for the intersection H of G with the kernel of much as we found the unipotent part of a triangularizable matrix group in Section 2.5.2. Use the algorithm in [Be1] to find a presentation for (G) and from that obtain a set fh 1 ; h l g of normal generators for H . Let e H = hh 1 ; h l i. Then H is the normal closure of e H in G. By conjugating elements in e H by g 1 ; g k , keep building up e H until e H is normal in G, in which ....

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R. Beals, Algorithms for matrix groups and the Tits alternative, Proc. 36th IEEE FOCS (1995), 593-602.

....directed increased attention to the development of algorithms for studying matrix groups. In particular, various recognition algorithms are of importance. These algorithms e#ciently identify a group given by a set of generators by deciding various properties of the group from the generators (cf. [1, 2, 3, 13, 14, 15] as well as the volumes [8, 9] and the many references therein) For a potentially infinite group, possibly the most fundamental property to be determined of a set of generators is that of finiteness. That is, given S # #, for an infinite group #, decide if #S#, the subgroup generated by S, is ....

R. Beals. Algorithms for matrix groups and the Tits alternative. In Proceedings of 36th Annual Symposium on Foundations of Computer Science, 1995, pp. 593-602.

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