Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the dimension of polynomial semirings   In our previous work, motivated by the study of tropical polynomials, a
definition for prime congruences was given for an arbitrary commutative
semiring. It was shown that for additively idempotent semirings this class
exhibits some analogous properties to prime ideals in ring theory. The current
paper focuses on the resulting notion of Krull dimension, which is defined as
the length of the longest chain of prime congruences. Our main result states
that for any additively idempotent semiring $A$, the semiring of polynomials
$A[x]$ and the semiring of Laurent polynomials $A(x)$, we have $\dim A[x] =
\dim A(x) = \dim A + 1$.

1
The categories of this paper are: cs
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat