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A statistical amoeba arises from a real-valued partition function when thepositivity condition for pre-exponential terms is relaxed, and families ofsignatures are taken into account. This notion lets us explore special types ofconstraints when we focus on those signatures that preserve particularproperties. Specifically, we look at sums of determinantal type, and mainattention is paid to a distinguished class of soliton solutions of theKadomtsev-Petviashvili (KP) II equation. A characterization of the signaturespreserving the determinantal form, as well as the signatures compatible withthe KP II equation, is provided: both of them are reduced to choices of signsfor columns and rows of a coefficient matrix, and they satisfy the whole KPhierarchy. Interpretations in term of information-theoretic properties,geometric characteristics, and the relation with tropical limits are discussed.

Signatures of partition functions and their complexity reduction through the KP II equation