In this paper, we investigate the generalized polynomial complementarity problem (GPCP), which is a subcase of the classic generalized complementarity problem (GCP) with the involved map pair being two polynomial maps; and meanwhile, it contains the recently studied polynomial complementarity problem (PCP) and the tensor complementarity problem (TCP) as special cases. With the help of the degree theory and two classes of structured tensor pairs: cone $R^{\mathbf{0}}$ tensor pair and cone $R^{\mathbf{d}}$ tensor pair, we obtain several results on the nonemptiness and compactness of the solution set of the GPCP. Specifically, when the leading tensor pair is cone $R^{\mathbf{0}}$, we establish two (topology) degree-theoretic results for the considered problem; and hence, we achieve two results which illustrate the connections between the solution set of the GPCP and the solution set of the TCP (or PCP), and all of them can reduce to the corresponding ones for PCPs and TCPs. When the leading tensor pair is cone $R^{\mathbf{d}}$, we show a theorem to describe this class of tensor pairs in terms of complementarity problems; and after that, we get two results which can reduce to the corresponding ones for PCPs and TCPs, and one broadenes the scope of the conditions in a very recent paper for the GPCP.