  Recently, H. Dao and R. Nair gave a combinatorial description of simplicial
complexes $\Delta$ such that the squarefree reduction of the Stanley-Reisner
ideal of $\Delta$ has the WLP in degree $1$ and characteristic zero. In this
paper, we apply the connections between analytic spread of equigenerated
monomial ideals, mixed multiplicities and birational monomial maps to give a
sufficient and necessary condition for the squarefree reduction $A(\Delta)$ to
satisfy the WLP in degree $i$ and characteristic zero in terms of mixed
multiplicities of monomial ideals that contain combinatorial information of
$\Delta$, we call them incidence ideals. As a consequence, we give an upper
bound to the possible failures of the WLP of $A(\Delta)$ in degree $i$ in
positive characteristics in terms of mixed multiplicities. Moreover, we extend
Dao and Nair's criterion to arbitrary monomial ideals in positive odd
characteristics.
