Two bivariate functions K,Q:Λ² -> F are said to be determinantally equivalent if for any n in N and x1,x2,...,xn in Λ, the determinants of matrices K(x_i,x_j) and Q(x_i,x_j) agree. We study to what extent such functions K and Q must be related by two canonical transformations corresponding to diagonal similarity and transposition.

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In the finite setting, this is closely related to the classical problem of Raphael Loewy concerning principal minors and diagonal similarity of matrices. While previous approaches rely heavily on linear-algebraic techniques, we present a fundamentally different and entirely combinatorial method.

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The key idea is to analyze permutations appearing in determinant expansions as cycles in a graph. In contrast to previous work in the nonvanishing setting, the presence of zeros requires substantially new combinatorial arguments, as the identities used in that setting are no longer available. From the resulting cycle structure, we construct two auxiliary functions whose cocycle properties encode the possible equivalence mechanisms between K and Q. The main step is then to prove a structural dichotomy for 3-cycles, showing that all such cycles necessarily fall into a single class. This ultimately yields a precise description of determinantally equivalent functions under a natural structural assumption.

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Time permitting, I will conclude with several open problems and possible directions for future research.

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Based on my recent preprint arXiv:2604.03934.