Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Affine Hilbert Functions of Unions of Layers in Finite Grids   The affine Hilbert function is a classical algebraic object that has been
central, among other tools, to the development of the polynomial method in
combinatorics. Owing to its concrete connections with Gr\"obner basis theory,
as well as its applicability in several areas like computational complexity,
combinatorial geometry, and coding theory, an important line of enquiry is to
understand the affine Hilbert function of structured sets of points in the
affine space.
  In this work, we determine the affine Hilbert function (over the reals) of
arbitrary unions of layers of points in a uniform grid (a finite grid with the
component sets having equispaced points), where each layer of points is
determined by a fixed sum of components for all the points. This extends a
result of Bernasconi and Egidi (Inf. Comput. 1999) from the Boolean cube
setting to the uniform grid setting.
  Our proofs follow a similar outline as that of Bernasconi and Egidi. However,
there are two bottlenecks that arise in the uniform grid setting. We resolve
these by using (i) a classical fact that a symmetric Jordan basis exists for
the function space on a uniform grid, and (ii) an extension to multisets of an
algebraic interpretation by Friedl and R\'onyai (Discrete Math. 2003) of the
notion of order shattering.
  The affine Hilbert function is, in fact, a stronger notion than the
finite-degree Zariski closure, which is yet another important tool in the
polynomial method toolkit. We conclude by giving an alternative proof of a
combinatorial characterization of a variant of finite-degree Zariski closures,
for unions of layers in uniform grids, obtained in an earlier work of the
author (arXiv, 2021).

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The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an