  We study the Hankel determinant generated by a Gaussian weight with
Fisher-Hartwig singularities of root type at $t_j$, $j=1,\cdots ,N$. It
characterizes a type of average characteristic polynomial of matrices from
Gaussian unitary ensembles. We derive the ladder operators satisfied by the
associated monic orthogonal polynomials and three compatibility conditions. By
using them and introducing $2N$ auxiliary quantities $\{R_{n,j}, r_{n,j},
j=1,\cdots,N\}$, we build a series of difference equations. Furthermore, we
prove that $\{R_{n,j}, r_{n,j}\}$ satisfy Riccati equations. From them we
deduce a system of second order PDEs satisfied by $\{R_{n,j}, j=1,\cdots,N\}$,
which reduces to a Painlev\'{e} IV equation for $N=1$. We also show that the
logarithmic derivative of the Hankel determinant satisfies the generalized
$\sigma$-form of a Painlev\'{e} IV equation.
