  Under a homogeneous magnetic field, we establish the local well-posedness of
low-regularity solutions for an ideal incompressible Magnetohydrodynamics (MHD)
system in Lagrangian coordinates. Firstly, we reduce the MHD system to a
degenerate wave-elliptic system inherent with a specific null form. After
constructing a new solution space, we can prove some good product estimates.
Combined with the inside null structure, a bilinear estimate of the
Klainerman-Machedon's type for nonlinear terms can be obtained. These lead us
to prove the local well-posedness of ideal incompressible MHD equations in
Lagrangian coordinates if the initial velocity $\bv_0 \in H^{s}(\mathbb{R}^n),
s> \frac{n+1}{2}$ $(n=2,3)$. So our result lowers $\frac12$-order regularity
comparing with the classical exponent $s>1+\frac{n}{2}$. Moreover, to the
author's knowledge, this is the first result concerning low-regularity
solutions of the ideal MHD equations.
