Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Uniform Version of the Petrov-Khovanskii Theorem   An Abelian integral is the integral over the level curves of a Hamiltonian
$H$ of an algebraic form $\omega$. The infinitesimal Hilbert sixteenth problem
calls for the study of the number of zeros of Abelian integrals in terms of the
degrees $H$ and $\omega$. Petrov and Khovanskii have shown that this number
grows at most linearly with the degree of $\omega$, but gave a purely
existential bound. Binyamini, Novikov and Yakovenko have given an
\emph{explicit} bound growing doubly-exponentially with the degree. We combine
the techniques used in the proofs of these two results, to obtain an explicit
bound on the number of zeros of Abelian integrals growing linearly with
$\deg\omega$.

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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph