Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-Level O-sequences of Codimension 3 and Degree of The Socle Elements   It is unknown if an Artinian level O-sequence of codimension 3 and type $r
(\ge 2)$ is unimodal, while it is known that any Gorenstein O-sequence of
codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence
of codimension 3 cannot be level. We also find another non-level case: if some
Artinian algebra $A$ of codimension 3 has the Hilbert function
$$\begin{matrix}\H & : & h_0 & h_1 & ... & h_{d-1} & \underbrace{h_d ...
h_d}_{s\text{-times}} & h_{d+s}, \end{matrix} $$ such that $h_d<h_{d+s}$ and
$s\ge 2$, then $A$ has a socle element in degree $d+s-2$, that is, $A$ is not
level.

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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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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: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga