  We present a new study of D semileptonic decays on the lattice which employs
the Highly Improved Staggered Quark (HISQ) action for both the charm and the
light valence quarks. We work with MILC unquenched $N_f = 2 + 1$ lattices and
determine the scalar form factor $f_0(q^2)$ for $D \rightarrow K, l \nu$
semileptonic decays. The form factor is obtained from a scalar current matrix
element that does not require any operator matching. We develop a new approach
to carrying out chiral/continuum extrapolations of $f_0(q^2)$. The method uses
the kinematic "$z$" variable instead of $q^2$ or the kaon energy $E_K$ and is
applicable over the entire physical $q^2$ range. We find $f^{D \rightarrow
K}_0(0) \equiv f^{D \rightarrow K}_+(0) = 0.747(19)$ in the chiral plus
continuum limit and hereby improve the theory error on this quantity by a
factor of $\sim$4 compared to previous lattice determinations. Combining the
new theory result with recent experimental measurements of the product $f^{D
\rightarrow K}_+(0) * |V_{cs}| $ from BaBar and CLEO-c leads to the most
precise direct determination of the CKM matrix element $|V_{cs}| $ to date,
$|V_{cs}| = 0.961(11)(24)$, where the first error comes from experiment and the
second is the lattice QCD theory error. We calculate the ratio $f^{D
\rightarrow K}_+(0)/f_{D_s}$ and find $2.986 \pm 0.087$ GeV$^{-1}$ and show
that this agrees with experiment.
