PURPOSE: To effect the selective polymerization of 1, 3-butadiene, isoprene or 1, 3-pentadiene, by polymerizing an unsaturated hydrocarbon mixture containing 1, 3-pentadiene in the presence of a polymerization catalyst containing a rare earth metal compound.

CONSTITUTION: Only 1, 3-butadiene, isoprene and 1,3-pentadiene are selectively polymerized by polymerizing a mixture containing at least one of 1,3-butadiene, isoprene and 1,3-pentadiene and other unsaturated 4W5C hydrocarbons in the presence of a polymerization catalyst containing a rare earth metal compound and trialkylaluminum of the formula: AlR3, wherein R is a 1W4C linear or branched alkyl. Said compounds include oxides, chlorides, phosphates, flurocarbonates, silicates, sulfates, nitrates, etc., of 17 elements including 15 elements from La of an atomic number 57 to Lu of an atomic number 71 and Sc and Y.