Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Finite sets of $d$-planes in affine space   Let $A$ be a subvariety of affine space $\mathbb{A}^n$ whose irreducible
components are $d$-dimensional linear or affine subspaces of $\mathbb{A}^n$.
Denote by $D(A)\subset\mathbb{N}^n$ the set of exponents of standard monomials
of $A$. We show that the combinatorial object $D(A)$ reflects the geometry of
$A$ in a very direct way. More precisely, we define a $d$-plane in
$\mathbb{N}^n$ as being a set $\gamma+\oplus_{j\in J}\mathbb{N}e_{j}$, where
$#J=d$ and $\gamma_{j}=0$ for all $j\in J$. We call the $d$-plane thus defined
to be parallel to $\oplus_{j\in J}\mathbb{N}e_{j}$. We show that the number of
$d$-planes in $D(A)$ equals the number of components of $A$. This generalises a
classical result, the finiteness algorithm, which holds in the case $d=0$. In
addition to that, we determine the number of all $d$-planes in $D(A)$ parallel
to $\oplus_{j\in J}\mathbb{N}e_{j}$, for all $J$. Furthermore, we describe
$D(A)$ in terms of the standard sets of the intersections
$A\cap\{X_{1}=\lambda\}$, where $\lambda$ runs through $\mathbb{A}^1$.

4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph