Skip Navigation



Journal of the London Mathematical Society Advance Access published online on November 20, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm070
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
76/3/777    most recent
jdm070v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Armstrong, S. N.
Right arrow Articles by Hillar, C. J.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2007 London Mathematical Society

Solvability of symmetric word equations in positive definite letters

Scott N. Armstrong

Department of Mathematics
University of California
Berkeley
CA 94720
USA
sarm{at}math.berkeley.edu

Christopher J. Hillar

Department of Mathematics
Texas A&M University
College Station
TX 77843
USA

Let S(X, B) be a symmetric (‘palindromic’) word in two letters X and B. A theorem due to Hillar and Johnson states that for each pair of positive definite matrices B and P, there is a positive definite solution X to the word equation S(X, B)=P. They also conjectured that these solutions are finite and unique. In this paper, we resolve a modified version of this conjecture by showing that the Brouwer degree of such an equation is equal to 1 (in the case of real matrices). It follows that, generically, the number of solutions is odd (and thus finite) in the real case. Our approach allows us to address the more subtle question of uniqueness by exhibiting equations with multiple real solutions, as well as providing a second proof of the result of Hillar and Johnson in the real case.

2000 Mathematics Subject Classification 15A24, 15A57; 15A18, 15A90.

The work of the second author is supported under a National Science Foundation Postdoctoral Fellowship.

Received July 27, 2006; revised March 16, 2007;
Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.