Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A stable version of Harbourne's Conjecture and the containment problem
  for space monomial curves   The symbolic powers $I^{(n)}$ of a radical ideal $I$ in a polynomial ring
consist of the functions that vanish up to order $n$ in the variety defined by
$I$. These do not necessarily coincide with the ordinary algebraic powers
$I^n$, but it is natural to compare the two notions. The containment problem
consists of determining the values of $n$ and $m$ for which $I^{(n)} \subseteq
I^m$ holds. When $I$ is an ideal of height $2$ in a regular ring, $I^{(3)}
\subseteq I^2$ may fail, but we show that this containment does hold for the
defining ideal of the space monomial curve $(t^a, t^b, t^c)$. More generally,
given a radical ideal $I$ of big height $h$, while the containment
$I^{(hn-h+1)} \subseteq I^n$ conjectured by Harbourne does not necessarily hold
for all $n$, we give sufficient conditions to guarantee such containments for
$n \gg 0$.

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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th