Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Range-compatible homomorphisms on spaces of symmetric or alternating
  matrices   Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field
$\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space
$\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S}
\rightarrow V$ is called range-compatible when it satisfies $F(s) \in
\mathrm{im}(s)$ for all $s \in \mathcal{S}$. Among the range-compatible maps
are the so-called local ones, that is the maps of the form $s \mapsto s(x)$ for
a fixed vector $x$ of $U$.
  In recent works, we have classified the range-compatible group homomorphisms
on $\mathcal{S}$ when the codimension of $\mathcal{S}$ in $\mathcal{L}(U,V)$ is
small. In the present article, we study the special case when $\mathcal{S}$ is
a linear subspace of the space $S_n(\mathbb{K})$ of all $n$ by $n$ symmetric
matrices: we prove that if the codimension of $\mathcal{S}$ in
$S_n(\mathbb{K})$ is less than or equal to $n-2$, then every range-compatible
homomorphism on $\mathcal{S}$ is local provided that $\mathbb{K}$ does not have
characteristic $2$. With the same assumption on the codimension of
$\mathcal{S}$, we also classify the range-compatible homomorphisms on
$\mathcal{S}$ when $\mathbb{K}$ has characteristic $2$. Finally, we prove that
if $\mathcal{S}$ is a linear subspace of the space $A_n(\mathbb{K})$ of all $n$
by $n$ alternating matrices with entries in $\mathbb{K}$, and the codimension
of $\mathcal{S}$ is less than or equal to $n-3$, then every range-compatible
homomorphism on $\mathcal{S}$ is local.

15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
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The categories of this paper are: adap-org