In this paper, we consider the minimization of two classes of polynomials over the standard simplex. These polynomials have their variables labeled by the edges of a complete uniform hypergraph, and their coefficients are defined in terms of some cardinality patterns of unions of edges. Data Envelopment Analysis (DEA) is a non-parametric method that aims to use scientific methods to investigate the performance of Decision-Making Units (DMUs). One of the interesting subjects in DEA is the minimization of the empirical error while satisfying some shape constraints, such as convexity and free disposability. In this research, the question is whether these polynomials attain their minimum value at the barycenter of the standard simplex, which corresponds to showing the optimality of the uniform distribution for the underlying queuing problem. The process focuses on the development of an adaptive observer-based Distributed Fault Estimation Observer (DFEO) for multi-agent nonlinear time-delay systems under a directed communication topology. The process involves constructing a fault estimation observer for each agent based on their relative output estimation errors.