Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Square-root cancellation for the signs of Latin squares   Let $L(n)$ be the number of Latin squares of order $n$, and let
$L^{\textrm{even}}(n)$ and $L^{\textrm{odd}}(n)$ be the number of even and odd
such squares, so that $L(n) = L^{\textrm{even}}(n) + L^{\textrm{odd}}(n)$. The
Alon-Tarsi conjecture states that $L^{\textrm{even}}(n)\neq
L^{\textrm{odd}}(n)$ when $n$ is even (when $n$ is odd the two are equal for
very simple reasons). In this short note we prove that $|L^{\textrm{even}}(n) -
L^{\textrm{odd}}(n)|\leq L(n)^{\frac{1}{2} + o(1)},$ thus establishing the
conjecture that the number of even and odd Latin squares, while conjecturally
not equal in even dimensions, are equal to leading order asymptotically. Two
proofs are given: both proceed by applying a differential operator to an
exponential integral over $\mathrm{SU}(n)$. The method is inspired by a recent
result of Kumar-Landsberg.

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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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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: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom