  We use the hard (Lipatov) pomeron for the low-x gluon distribution and
provide a compact formula for the ratio $R^{c} =F_{L}^{c}/F_{2}^{c}$ that is
useful to extract the charm structure function from the reduced charm
cross-section, in particular at DESY HERA. Our results show that this ratio is
independent of x and independent of the DGLAP evolution of the gluon PDF. As a
result, we show that the charm structure function and the reduced charm
cross-section exponents do not have the same behaviour at very low x. This
difference is independent of the input gluon distribution functions and
predicts the non-linear effects and some evidence for shadowing and
antishadowing at HERA and RHIC.
