In this paper, the polynomial hypergraph filter (PHF) designs using least squares, minimax and peak constrained methods are presented. First, the hypergraph Laplacian matrix (HLM) is used to define the hypergraph Fourier transform (HFT) by using the eigen-decomposition of HLM. Then, the transfer matrix and spectral response of PHF are defined by using HLM and HFT. Next, the filter coefficients of PHF are determined by minimizing three error measures between actual response and ideal response including least squares (LS) error, maximum error, and LS error with peak constraint. Finally, the distributed implementation structure of PHF is studied and noise reduction application example is demonstrated to show the effectiveness of the proposed methods.