Topology optimization is among the most effective tools for innovative and lightweight structural designs. Multi-material design is considered to achieve better structural performance than single-material design. To significantly reduce the design space dimensionality and facilitate the optimization of multi-material structural design problems, this study proposes an effective topological representation and dimensionality reduction approach based on the material-field series expansion (MFSE) model. In the proposed method, a specified number of material phases is described within a single material field with a piecewise Heaviside projection function. The topology optimization problem is solved by determining the optimal MFSE coefficients. Owing to the single material-field topological description and series expansion, the number of design variables is independent of the finite element mesh and the number of material phases. In terms of dimensionality reduction, the proposed method outperformed all reported state-of-the-art algorithms for multi-material topology optimization. The validity and universality of the proposed method are illustrated in two- and three-dimensional numerical examples.