The present invention relates to diene polymers or diene copolymers, wherein the diene polymers or diene copolymers have, at the start of the polymer chains, tertiary amino groups of the formula (Ia), (Ib), (IIa) or (IIb) where R1, R2 are the same or different and are each alkyl, cycloalkyl, aryl, alkaryl and aralkyl radicals which may contain heteroatoms such as O, N, S and/or Si, Z is a divalent organic radical which, as well as C and H, may contain heteroatoms such as O, N, S and/or Si, and, at the end of the polymer chains, silane-containing carbinol groups of the formula (III) or metal salts thereof or semimetal salts thereof, where R3, R4, R5, R6 are the same or different and are each an H or alkyl, cycloalkyl, aryl, alkaryl and aralkyl radicals which may contain heteroatoms such as O, N, S and/or Si, A is a divalent organic radical which, as well as C and H, may contain heteroatoms such as O, N, S and/or Si.