  In this paper, we establish local well-posedness for the Zakharov system on
$\mathbb{T}^d$, $d\ge3$ in a low regularity setting. Our result improves the
work of Kishimoto. Moreover, the result is sharp up to $\varepsilon$-loss of
regularity when $d=3$ and $d\ge5$ as long as one utilizes the iteration
argument. We introduce ideas from recent developments of the Fourier
restriction theory. The key element in the proof of our well-posedness result
is a new trilinear discrete Fourier restriction estimate involving paraboloid
and cone. We prove this trilinear estimate by improving Bourgain--Demeter's
range of exponent for the linear decoupling inequality for paraboloid under the
constraint that the input space-time function $f$ satisfies ${\rm supp}\,
\hat{f} \subset \{ (\xi,\tau) \in \mathbb{R}^{d+1}: 1- \frac1N \le |\xi| \le 1
+ \frac1N,\; |\tau - |\xi|^2| \le \frac1{N^{2}} \} $ for large $N\ge1$.
