  Let $\mathcal{A}$ be a central hyperplane arrangement in $\mathbb{C}^{n+1}$
and $H_i,i=1,2,...,d$ be the defining equations of the hyperplanes of
$\mathcal{A}$. Let $f=\prod_i H_i$. There is a global Milnor fibration
$F\hookrightarrow \mathbb{C}^{n+1} \setminus \mathcal{A} \xrightarrow{f}
\mathbb{C}^*,$ where $F$ is called the Milnor fiber and can be identified as
the affine hypersurface $f=1$ in $\mathbb{C}^{n+1}$.
  Many open questions have been raised subject to $F$. In particular, it has
been conjectured that the integral homology, or the characteristic polynomial,
hence the Betti numbers, of $F$ are also determined by the intersection lattice
$L(\mathcal{A})$.
  In this paper, we find a combinatorial upper bound for the first the
characteristic polynomial of the Milnor fiber for central hyperplane
arrangements, which improves existing results in the study. As a corollary, we
obtain a combinatorial obstruction for trivial algebraic monodromy of the first
homology of Milnor fiber. Calculations and comparisons to known examples
computed using Fox calculus will be provided.

Geometric Topology