TY - JOUR

T1 - Solvability of symmetric word equations in positive definite letters

AU - Armstrong, Scott N.

AU - Hillar, Christopher J.

N1 - Funding Information:
Received 27 July 2006; revised 16 March 2007; published online 20 November 2007. 2000 Mathematics Subject Classification 15A24, 15A57; 15A18, 15A90. The work of the second author is supported under a National Science Foundation Postdoctoral Fellowship.

PY - 2007/12

Y1 - 2007/12

N2 - 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.

AB - 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.

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U2 - 10.1112/jlms/jdm070

DO - 10.1112/jlms/jdm070

M3 - Article

AN - SCOPUS:44649180514

SN - 0024-6107

VL - 76

SP - 777

EP - 796

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 3

ER -