  We study inequalities connecting a product of uniform norms of polynomials
with the norm of their product. Generalizing Gel'fond-Mahler inequality for the
unit disk and Kneser-Borwein inequality for the segment $[-1,1]$, we prove an
asymptotically sharp inequality for norms of products of algebraic polynomials
over an arbitrary compact set in plane. Applying similar techniques, we produce
a related inequality for the norm of a single monic factor of a monic
polynomial. The best constants in both inequalities are obtained by potential
theoretic methods. We also consider applications of the general results to the
cases of a disk and a segment.
