We present exact solutions for the zero-temperature partition function
(chromatic polynomial $P$) and the ground state degeneracy per site $W$ (=
exponent of the ground-state entropy) for the $q$-state Potts antiferromagnet
on strips of the square lattice of width $L_y$ vertices and arbitrarily great
length $L_x$ vertices. The specific solutions are for (a) $L_y=4$,
$(FBC_y,PBC_x)$ (cyclic); (b) $L_y=4$, $(FBC_y,TPBC_x)$ (M\"obius); (c)
$L_y=5,6$, $(PBC_y,FBC_x)$ (cylindrical); and (d) $L_y=5$, $(FBC_y,FBC_x)$
(open), where $FBC$, $PBC$, and $TPBC$ denote free, periodic, and twisted
periodic boundary conditions, respectively. In the $L_x \to \infty$ limit of
each strip we discuss the analytic structure of $W$ in the complex $q$ plane.
The respective $W$ functions are evaluated numerically for various values of
$q$. Several inferences are presented for the chromatic polynomials and
analytic structure of $W$ for lattice strips with arbitrarily great $L_y$. The
absence of a nonpathological $L_x \to \infty$ limit for real nonintegral $q$ in
the interval $0 < q < 3$ ($0 < q < 4$) for strips of the square (triangular)
lattice is discussed.