Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dyck path triangulations and extendability   We introduce the Dyck path triangulation of the cartesian product of two
simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever $m\geq k>n$, any triangulation of
$\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of
$\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we
prove that the bound $k>n$ is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.

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The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
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The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn