  In recent algorithms that use deformation in order to compute the number of
points on varieties over a finite field, certain differential equations of
matrices over p-adic fields emerge. We present a novel strategy to solve this
kind of equations in a memory efficient way. The main application is an
algorithm requiring quasi-cubic time and only quadratic memory in the parameter
n, that solves the following problem: for E a hyperelliptic curve of genus g
over a finite field of extension degree n and small characteristic, compute its
zeta function. This improves substantially upon Kedlaya's result which has the
same quasi-cubic time asymptotic, but requires also cubic memory size.
