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

The predictions of the charm structure function exponents behaviour at low x in deep inelastic scattering