自由模的实零点定理
A Real Nullstellensatz for Free Modules
摘要：
令$A$是所有$n \times n$矩阵的代数，其元素来自$\RR[x_1,\ldots,x_d]$，令$G_1,\ldots,G_m,F \in A$。我们将证明对于每个$a \in \RR^d$和$v \in \RR^n$的$F(a)v=0$使得对于所有$i$的$G_i(a)v=0$当且仅当$F$属于包含$G_1,\ldots,G_m$的$A$的最小实左理想。这里$A$的左理想$J$是实的，如果对于每个$H_1,\ldots,H_k \in A$使得$H_1^T H_1+\ldots+H_k^T H_k \in J+J^T$，我们有那个$H_1,\ldots,H_k \in J$。我们称这个结果为矩阵多项式的单侧实零点。我们首先证明了归纳$n$，它成立时，$G_1,\ldots,G_m,F$有零除了在第一行。这个辅助结果可以用公式表示为自由模$\RR[x_1,\ldots,x_d]^n$的实零点。
英文摘要：
Let $A$ be the algebra of all $n \times n$ matrices with entries from $\RR[x_1,\ldots,x_d]$ and let $G_1,\ldots,G_m,F \in A$. We will show that $F(a)v=0$ for every $a \in \RR^d$ and $v \in \RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\ldots,H_k \in A$ such that $H_1^T H_1+\ldots+H_k^T H_k \in J+J^T$ we have that $H_1,\ldots,H_k \in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n$ that it holds when $G_1,\ldots,G_m,F$ have zeros everywhere except in the first row. This auxiliary result can be formulated as a Real Nullstellensatz for the free module $\RR[x_1,\ldots,x_d]^n$.
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