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The uncertainties in estimating the hadronic production of the $B_c$ mesonare studied under the framework of the complete $\alpha_s^4$ approach of theperturbative QCD and the gluon-gluon fusion mechanism. Quantitative comparisonson the production at TEVATRON and LHC are made. Considering the detectors atTEVATRON and LHC, we have also estimated the production with proper kinematiccuts. Based on the results, we conclude that the experimental studies of the$B_c$ meson at the two colliders will be complimentary and stimulative. We findthat as c.m. energy is increasing from RUN-I to RUN-II at TEVATRON, theproduction cross-section increases about 20%.Comment: 22 page, 11 figures, for the revised version the contents (figures  and text) are revised and a few references are adde

Uncertainties in estimating B c hadronic productionand comparisons of the production at TEVATRON and LHC