Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Real Nullstellensatz for Free Modules   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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The categories of this paper are: plasm-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex