Let $A$ be a $n\times n$ real matrix. The piecewise linear equation system $z-A\vert z\vert =b$ is called an absolute value equation (AVE). It is well-known to be equivalent to the linear complementarity problem. Unique solvability of the AVE is known to be characterized in terms of a generalized Perron root called the sign-real spectral radius of $A$. For mere, possibly non-unique, solvability no such characterization exists. We narrow this gap in the theory. That is, we define the concept of the aligned spectrum of $A$ and prove, under some mild genericity assumptions on $A$, that the mapping degree of the piecewise linear function $F_A:\mathbb{R}^n\to\mathbb{R}^n\,, z\mapsto z-A\lvert z\rvert$ is congruent to $(k+1)\mod 2$, where $k$ is the number of aligned values of $A$ which are larger than $1$. We also derive an exact--but more technical--formula for the degree of $F_A$ in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP.

翻译：设$A$为$n\times n$阶实矩阵。分段线性方程组$z-A\vert z\vert =b$称为绝对值方程（AVE），它等价于线性互补问题。已知AVE的唯一可解性可通过一个称为$A$的符号实谱半径的广义Perron根来刻画。但对于一般可能非唯一的可解性，尚无此类刻画。本文填补了这一理论空白：我们定义了$A$的对齐谱概念，并在$A$满足某些温和泛化性假设的条件下，证明了分段线性函数$F_A:\mathbb{R}^n\to\mathbb{R}^n\,， z\mapsto z-A\lvert z\rvert$的映射度同余于$(k+1)\mod 2$，其中$k$是$A$的大于$1$的对齐值个数。此外，我们推导了以对齐谱表达的$F_A$度的精确（但偏技术性）公式。最后，我们给出了线性互补问题的对应量及结果。