Nonsymmetric generic matrix equations

Abstract

Let $(A_i)_{0\leq i\leq k}$ 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 $A_kX^k+\cdots+A_1X+A_0=0_n$ has $\binom{nk}{n}$ 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 $(B_i)_{i\leq k}$ 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 $X^k+C_{k-1}X^{k-1}+\cdots+C_1X+C_0=0_n$ 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 $A_1XA_2X+XA_3X+X^2A_4+A_5X+A_6=0_2$ admits $16$ isolated solutions in $\mathcal{M}_2(\overline{K})$, that is, according to the B\'ezout's theorem, the maximum for a quadratic $2\times 2$ matrix equation.