The invention relates to thermoplastic compositions comprising a polymer matrix and an additive which modifies the rheological behavior of the matrix in the molten state. The purpose of the invention is to provide a preferably non-reactive additive which can be dispersed in the matrix and which can be used to obtain a good compromise in terms of rheological properties/mechanical properties. According to the invention, the additive is a hyperbranched polymer which is functionalized by R<2> radicals, R<2> being a radical of the following type: substituted or non-substituted hydrocarbon, of the silicon type, linear or branched alkyl, aromatic, arylalkyl, alkylaryl or cycloaliphatic, which can comprise one or more unsaturations and/or one or more heteroatoms. Preferably, the composition does not contain hyperbranched polymers which produce therein a reduction in the molar mass of matrix M which is greater than or equal to 7% in relation to a sample composition which comprises matrix M but which does not contain hyperbranched polymer additive(s). Said mass measurement is taken preferably using a predetermined protocol P. The invention is suitable for molding, extrusion, engineered plastics, wires and fibers.