Determinantally equivalent nonzero functions

We study the problem raised in Marco Stevens (2021) [20] concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples. The problem is plainly stated as follows: Let Λ be a set and F a field, and suppose that K , Q : Λ 2 → F are two functions such that for any n ∈ N and x 1 , x 2 , … , x n ∈ Λ , the determinants of matrices ( K ( x i , x j ) ) 1 ≤ i , j ≤ n and ( Q ( x i , x j ) ) 1 ≤ i , j ≤ n agree. What are all the possible transformations that transform Q into K ? In Marco Stevens (2021) [20] the following two were conjectured: ( T f ) ( x , y ) = f ( y , x ) ; and ( T f ) ( x , y ) = g ( x ) g ( y ) − 1 f ( x , y ) for some nowhere-zero function g . In the same paper, this conjectured classification is verified in the case of symmetric functions K and Q . By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed. By taking Λ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23–64]. Hence, our paper presents a simple non-linear-algebraic proof that uses only some elementary combinatorics and three simple algebraic identities involving 3-cycles and 4-cycles.