Nonsymmetric generic matrix equations
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 $A_kX^k+\cdots+A_1X+A_0=0_n$ h...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.1304.2506 |