  We solve here the so called division problem for wave equations with generic
quadratic non-linearities in high dimensions. Specifically, we show that
semilinear wave equations which can be written as systems involving quadratic
derivative non-linearities are globally well posed in (6+1) and higher
dimensions for all regularities greater than the scaling. This paper is the
first in a series of works where we discuss the global regularity properties of
general non-linear wave equations for all spatial dimensions greater than or
equal to 4.

Analysis of PDEs