Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A probabilistic heuristic for counting components of functional graphs
  of polynomials over finite fields   In 2014, Flynn and the second author bounded the average number of components
of the functional graphs of polynomials of fixed degree over a finite field.
When the fixed degree was large (relative to the size of the finite field),
their lower bound matched Kruskal's asymptotic for random functional graphs.
However, when the fixed degree was small, they were unable to match Krusal's
bound, since they could not (Lagrange) interpolate cycles in functional graphs
of length greater than the fixed degree. In our work, we introduce a heuristic
for approximating the average number of such cycles of any length. This
heuristic is, roughly, that for sets of edges in a functional graph, the
quality of being a cycle and the quality of being interpolable are
"uncorrelated enough". We prove that this heuristic implies that the average
number of components of the functional graphs of polynomials of fixed degree
over a finite field is within a bounded constant of Kruskal's bound. We also
analyze some numerical data comparing implications of this heuristic to some
component counts of functional graphs of polynomials over finite fields.

5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an