Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Additive bases, coset covers, and non-vanishing linear maps   Recently, the first two authors proved the Alon-Jaeger-Tarsi conjecture on
non-vanishing linear maps, for large primes. We extend their ideas to address
several other related conjectures.
  We prove the weak Additive Basis conjecture proposed by Szegedy, making a
significant step towards the Additive Basis conjecture of Jaeger, Linial,
Payan, and Tarsi. In fact, we prove it in a strong form: there exists a set
$A\subset\mathbb{F}_p^*$ of size $O(\log p)$ such that if
$B\subset\mathbb{F}_p^{n}$ is the union of $p$ linear bases, then $A\cdot
B=\{a\cdot v:a\in A, v\in B\}$ is an additive basis.
  An old result of Tomkinson states that if $G$ is a group, and
$\{H_{i}x_{i}:i\in [k]\}$ is an irredundant coset cover of $G$, then
$|G:\bigcap_{i\in [k]} H_{i}|\leq k!,$ and this bound is the best possible. It
is a longstanding open problem whether the upper bound can be improved to
$e^{O(k)}$ in case we restrict cosets to subgroups. Pyber proposed to study
this question for abelian groups. We show that somewhat surprisingly, if $G$ is
abelian, the upper bound can be improved to $e^{O(k\log \log k)}$ already in
the case of general coset covers, making the first substantial improvement over
the $k!$ bound.
  Finally, we prove a natural generalization of the Alon-Jaeger-Tarsi
conjecture for multiple matrices.

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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin