Universal bound on the cardinality of local hidden variables in networks

We present an algebraic description of the sets of local correlations in arbitrary networks, when the parties have finite inputs and outputs. We consider networks generalizing the usual Bell scenarios by the presence of multiple uncorrelated sources. We prove a finite upper bound on the cardinality of the value sets of the local hidden variables. Consequently, we find that the sets of local correlations are connected, closed and semialgebraic, and bounded by tight polynomial Bell-like inequalities.