Complexity reduction for sign configurations through the KP II equation and its information-theoretic aspects

We provide a combinatorial setting to explore the information content associated with the fulfillment of the Kadomtsev-Petviashvili (KP) II equation. We start from a special family of solutions of KP II, namely, tau-functions of Wronskian type. A solution in this class can be expressed as a sum of exponentials, and we look at combinations of signs (signatures) for the nonvanishing terms in this sum. We prove a characterization of the signatures that return another solution of the KP II equation: they can be represented as choices of signs for columns and rows of a coefficient matrix, so we recover a function satisfying the whole KP hierarchy from this single constraint. The redundancy of this representation for different choices of the initial solution is investigated. Enumerative, information-theoretic, and geometric aspects of this construction are discussed.