Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polynomials and harmonic functions on discrete groups   Alexopoulos proved that on a finitely generated virtually nilpotent group,
the restriction of a harmonic function of polynomial growth to a torsion-free
nilpotent subgroup of finite index is always a polynomial in the Mal'cev
coordinates of that subgroup. For general groups, vanishing of higher-order
discrete derivatives gives a natural notion of polynomial maps, which has been
considered by Leibman and others. We provide a simple proof of Alexopoulos's
result using this notion of polynomials, under the weaker hypothesis that the
space of harmonic functions of polynomial growth of degree at most k is finite
dimensional. We also prove that for a finitely generated group the Laplacian
maps the polynomials of degree k surjectively onto the polynomials of degree k
- 2. We then present some corollaries. In particular, we calculate precisely
the dimension of the space of harmonic functions of polynomial growth of degree
at most k on a virtually nilpotent group, extending an old result of Heilbronn
for the abelian case, and refining a more recent result of Hua and Jost.

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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: cs
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: eess