Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Degree of Regularity of Linear Homogeneous Equations   We define a linear homogeneous equation to be strongly r-regular if, when a
finite number of inequalities is added to the equation, the system of the
equation and inequalities is still r-regular. In this paper, we show that, if a
linear homogeneous equation is r-regular, then it is strongly r-regular. In
2009, Alexeev and Tsimerman introduced a family of equations, each of which is
(n-1)-regular but not n-regular, verifying a conjecture of Rado from 1933.
These equations are actually strongly (n-1)-regular as an immediate corollary
of our results.

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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph