Solvability of symmetric word equations in positive definite letters

Scott N. Armstrong, Christopher J. Hillar

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)777-796
Number of pages20
JournalJournal of the London Mathematical Society
Volume76
Issue number3
DOIs
StatePublished - Dec 2007

ASJC Scopus subject areas

  • Mathematics(all)

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