Nonsymmetric generic matrix equations

TL;DR

This paper investigates the solutions of various nonsymmetric matrix equations over fields, revealing their number, structure, and properties, including generic solutions, commuting solutions, and maximum solutions for specific cases.

Contribution

It provides new results on the number and nature of solutions for generic nonsymmetric matrix equations, including explicit counts and conditions for commutativity and simplicity.

Findings

The generic unilateral matrix equation has inom{nk}{n} solutions over an algebraic closure.

For fixed matrices, the solutions to certain polynomial equations are finite and simple.

When n=2, specific quadratic matrix equations have 16 solutions, reaching the maximum predicted by Bézout's theorem.

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.