Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniform Bounds in F-Finite Rings and Lower Semi-Continuity of the
  F-Signature   This paper establishes uniform bounds in characteristic $p$ rings which are
either F-finite or essentially of finite type over an excellent local ring.
These uniform bounds are then used to show that the Hilbert-Kunz length
functions and the normalized Frobenius splitting numbers defined on the
Spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz
multiplicity function and the F-signature function. From this we establish that
the F-signature function is lower semi-continuous. We also give a new proof of
the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally
proven by Ilya Smirnov.

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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex