Method for producing polymerization catalyst composition, polymerization catalyst composition, met

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A method for producing a polymer containing silica that does not involve long kneading time, and a polymer composition having an intended low heat generation property. A polymerization catalyst composition is produced by mixing and aging a second element and a third element, and then adding a first element to the mixture to react the first element with the mixture. The first element contains a compound that contains a rare earth metal element, the second element contains a compound represented by the following formula (X), and the third element contains silica. YR1aR2bR3c (X) (In the formula, Y is a metal; R1 and R2 are hydrogen atoms or hydrocarbon groups; and R3 is a hydrocarbon group, and R1, R2, and R3 are the same as or different from each other, a, b, and c are 0 or 1.).