Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rationality of Equivariant Hilbert Series and Asymptotic Properties   An FI- or an OI-module $\mathbf{M}$ over a corresponding noetherian
polynomial algebra $\mathbf{P}$ may be thought of as a sequence of compatible
modules $\mathbf{M}_n$ over a polynomial ring $\mathbf{P}_n$ whose number of
variables depends linearly on $n$. In order to study invariants of the modules
$\mathbf{M}_n$ in dependence of $n$, an equivariant Hilbert series is
introduced if $\mathbf{M}$ is graded. If $\mathbf{M}$ is also finitely
generated, it is shown that this series is a rational function. Moreover, if
this function is written in reduced form rather precise information about the
irreducible factors of the denominator is obtained. This is key for
applications. It follows that the Krull dimension of the modules $\mathbf{M}_n$
grows eventually linearly in $n$, whereas the multiplicity of $\mathbf{M}_n$
grows eventually exponentially in $n$. Moreover, for any fixed degree $j$, the
vector space dimensions of the degree $j$ components of $\mathbf{M}_n$ grow
eventually polynomially in $n$. As a consequence, any graded Betti number of
$\mathbf{M}_n$ in a fixed homological degree and a fixed internal degree grows
eventually polynomially in $n$. Furthermore, evidence is obtained to support a
conjecture that the Castelnuovo-Mumford regularity and the projective dimension
of $\mathbf{M}_n$ both grow eventually linearly in $n$. It is also shown that
modules $\mathbf{M}$ whose width $n$ components $\mathbf{M}_n$ are eventually
Artinian can be characterized by their equivariant Hilbert series. Using
regular languages and finite automata, an algorithm for computing equivariant
Hilbert series is presented.

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The categories of this paper are: hep-th
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn