正定对称字方程的可解性
Solvability of Symmetric Word Equations in Positive Definite Letters
摘要：
设$S(X,B)$是一个对称（“回文”）字，有两个字母$X$和$B$。Hillar和Johnson的一个定理指出，对于每对正定矩阵$B$和$P$，存在字方程$S(X,B)=P$的正定解$X$。他们还强调，这些解决方案是有限的和唯一的。在本文中，我们解决了这个猜想的一个修改版本，证明了这样的方程的Brouwer度等于1（在实矩阵的情况下）。因此，一般来说，在真实情况下，解的数量是奇数（因此是有限的）。我们的方法使我们能够解决更微妙的问题的唯一性，展示方程的多个真正的解决方案，以及提供第二个证明的结果，希拉尔和约翰逊在真实的情况下。
英文摘要：
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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