Nonsymmetric generic matrix equations

Abstract

Let $(A_i{)}_{0\le i\le k}$ be generic matrices over $\mathbb{Q}$, the field of rational numbers. Let $K=\mathbb{Q}(E)$, where $E$ denotes the entries of the $(A_i{)}_i$, and let $\overline{K}$ be the algebraic closure of $K$. We show that the generic unilateral equation $A_k{X}^k+\cdots +A_{1}X+A_0=0_n$ has $\left(\genfrac{}{}{0ex}{}{nk}{n}\right)$ solutions $X\in {\mathcal{M}}_n(\overline{K})$. Solving the previous equation is equivalent to solving a polynomial of degree $kn$, with Galois group $S_{kn}$ over $K$. Let $(B_i{)}_{i\le k}$ be fixed $n\times n$ matrices with entries in a field $L$. We show that, for a generic $C\in {\mathcal{M}}_n(L)$, a polynomial equation $g(B_1,\cdots ,B_k,X)=C$ admits a finite fixed number of solutions and these solutions are simple. We study, when $n=2$, the generic non-unilateral equations $X^2+BXC+D=0_2$ and $X^2+BXB+C=0_2$. We consider the unilateral equation $X^k+C_{k-1}{X}^{k-1}+\cdots +C_{1}X+C_0=0_n$ when the $(C_i{)}_i$ are $n\times n$ generic commuting matrices ; we show that every solution $X\in {\mathcal{M}}_n(\overline{K})$ commutes with the $(C_i{)}_i$. When $n=2$, we prove that the generic equation $A_{1}X{A}_{2}X+X{A}_{3}X+X^2{A}_4+A_{5}X+A_6=0_2$ admits $16$ isolated solutions in ${\mathcal{M}}_2(\overline{K})$, that is, according to the Bézout's theorem, the maximum for a quadratic $2\times 2$ matrix equation.

19 pages