The invention is directed to essentially saturated hydrocarbon polymer composition comprising essentially saturated hydrocarbon polymers having A) a backbone chain, B) a plurality of essentially hydrocarbyl sidechains connected to A), said sidechains each having a number-average molecular weight of from 2500 Daltons to 125,000 Daltons and a MWD by SEC of 1.0-3.5; and having A) a Newtonian limiting viscosity (.eta..sub.0) at 190.degree. C. at least 50% greater than that of a linear olefinic polymer of the same chemical composition and weight average molecular weight, preferably at least twice as great as that of said linear polymer, B) a ratio of the rubbery plateau modulus at 190.degree. C. to that of a linear polymer of the same chemical composition less than 0.5, preferably <0.3, C) a ratio of the Newtonian limiting viscosity (.eta..sub.0) to the absolute value of the complex viscosity in oscillatory shear (.eta.*)at 100 rad/sec at 190.degree. C. of at least 5, and D) a ratio of the extensional viscosity measured at a strain rate of 1 sec.sup.-1, 190.degree. C., and time=3 sec (i.e., a strain of 3) to that predicted by linear viscoelasticity at the same temperature and time of 2 or greater. Ethylene-butene prepared by anionic polymerization and hydrogenation illustrate and ethylene-hexene copolymers prepared by coordination polymerization illustrate the invention. The invention polymers exhibit improved processing characteristics in that the shear thinning behavior closely approaches that of ideal polymers and exhibit improved strain thickening.