  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.
