Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T17:16:36.609Z Has data issue: false hasContentIssue false
  • English
  • Français

On Tensor-Factorisation Problems,I: The Combinatorial Problem

Published online by Cambridge University Press:  01 February 2010

Peter M. Neumann  and
Cheryl E. Praeger
Peter M. Neumann
Affiliation:
The Queen's College, Oxford 0X1 4AW, United Kingdom, [email protected]
Cheryl E. Praeger
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia, [email protected], http://www.maths.uwa.edu.au/~praeger

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A k-multiset is an unordered k-tuple, perhaps with repetitions. If x is an r-multiset {x1, …, xr} and y is an s-multiset {y1, …, ys} with elements from an abelian group A the tensor product x ⊗ y is defined as the rs-multiset {xi yj | 1 ≤ i ≤ r, 1 ≤ j ≤ s}. The main focus of this paper is a polynomial-time algorithm to discover whether a given rs-multiset from A can be factorised. The algorithm is not guaranteed to succeed, but there is an acceptably small upper bound for the probability of failure. The paper also contains a description of the context of this factorisation problem, and the beginnings of an attack on the following division-problem: is a given rs-multiset divisible by a given r-multiset, and if so, how can division be achieved in polynomially bounded time?

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 7 , 2004 , pp. 73 - 100
Copyright
Copyright © London Mathematical Society 2004

References

Referances

1. Aho, Alfred V., Hopcroft, John E., Ullman, Jeffrey D., Data structures and algorithms (Addison-Wesley, Reading, MA, 1983).Google Scholar
2. Arratia, Richard, Barbour, A. D. and Tavaré, Simon, ‘On random polynomials over finite fields’, Math. Proc. Cambridge Philos. Soc. 114 (1993) 347–368.CrossRefGoogle Scholar
3. Greenhill, Catherine, ‘From multisets to matrix groups: some algorithms related to the exterior square’ DPhil thesis, Oxford, (1996).Google Scholar
4. Greenhill, Catherine, ‘An algorithm for recognising the exterior square of a matrix’, Linear and Multilinear Algebra 46 (1999) 213–244.CrossRefGoogle Scholar
5. Greenhill, Catherine, ‘An algorithm for recognising the exterior square of a multiset’, LMS J. Comput. Math. 3 (2000) 96–116; http://www.lms.ac.Uk/jcm/3/lmsl999-021.CrossRefGoogle Scholar
6. Huang, Jia Lun, ‘The implementation of a factorisation algorithm for combinatorial tensor products’, MSc thesis, Oxford, August (2003).Google Scholar
7. Jacobson, Nathan, Lectures in abstract algebra, vols I–III (Van Nostrand, New Jersey, 1953).CrossRefGoogle Scholar
8. Knuth, Donald E., The art of computer programming, vol 3: ‘Sorting and searching’ (Addison-Wesley, Reading, MA, 1973).Google Scholar
9. Mignotte, M. and Nicolas, J.-L, ‘Statistique sur Fq [X]’, Ann. Inst. H. Poincaré Sect. B (N.S.) 19 (1983) 113–121.Google Scholar