  For linear recurrence systems, the problem of finding rational solutions is
reduced to the problem of computing polynomial solutions by computing a content
bound or a denominator bound. There are several bounds in the literature. The
sharpest bound leads to polynomial solutions of lower degrees, but this
advantage need not compensate for the time spent on computing that bound.
  To strike the best balance between sharpness of the bound versus CPU time
spent obtaining it, we will give a family of bounds. The $J$'th member of this
family is similar to (Abramov, Barkatou, 1998) when $J=1$, similar to (van
Hoeij, 1998) when $J$ is large, and novel for intermediate values of $J$, which
give the best balance between sharpness and CPU time.
  The setting for our content bounds are systems $\tau(Y) = MY$ where $\tau$ is
an automorphism of a UFD, and $M$ is an invertible matrix with entries in its
field of fractions. This setting includes the shift case, the $q$-shift case,
the multi-basic case and others. We give two versions, a global version, and a
version that bounds each entry separately.
