Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized Dissections and Monsky's Theorem   Monsky's celebrated equidissection theorem follows from his more general
proof of the existence of a polynomial relation $f$ among the areas of the
triangles in a dissection of the unit square. More recently, the authors
studied a different polynomial $p$, also a relation among the areas of the
triangles in such a dissection, that is invariant under certain deformations of
the dissection. In this paper we study the relationship between these two
polynomials.
  We first generalize the notion of dissection, allowing triangles whose
orientation differs from that of the plane. We define a deformation space of
these generalized dissections and we show that this space is an irreducible
algebraic variety. We then extend the theorem of Monsky to the context of
generalized dissections, showing that Monsky's polynomial $f$ can be chosen to
be invariant under deformation. Although $f$ is not uniquely defined, the
interplay between $p$ and $f$ then allows us to identify a canonical pair of
choices for the polynomial $f$. In many cases, all of the coefficients of the
canonical $f$ polynomials are positive. We also use the deformation-invariance
of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the
sum of its variables.

20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con