Determinantally Equivalent Functions Beyond the Nowhere-Zero Case
Harry Sapranidis Mantelos
Abstract
Let $Λ$ be a set and $\mathbb{F}$ a field, and suppose that $K,Q:Λ^2\to\mathbb{F}$ are two functions such that for any $n\in\mathbb{N}$ and $x_1,x_2,\ldots,x_n\inΛ$, the determinants of matrices $(K(x_i,x_j))_{1\leq i,j\leq n}$ and $(Q(x_i,x_j))_{1\leq i,j\leq n}$ agree. To what extent is it true that solely the two canonical transformations $(Tf)(x,y)=f(y,x)$ and $(Tf)(x,y)=g(x)g(y)^{-1}f(x,y)$, for some nowhere-zero function $g$, can be used to transform $Q$ into $K$? In the symmetric case, this holds without further assumptions (see [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021]). Without symmetry, however, the statement fails in general. In [Harry Sapranidis Mantelos, Determinantally equivalent nonzero functions, Discrete Mathematics, 349(6):115021, 2026], it is shown that the conclusion remains valid under a natural structural condition, referred to as property D, together with the additional assumption that both functions are nowhere zero. In the present paper, we remove this nowhere-zero hypothesis. Building on the combinatorial framework introduced in Mantelos (2026), we extend its underlying principle through a detailed analysis of the implications of property D. Our proof avoids linear algebra entirely and instead exploits the combinatorial structure of permutations in the definition of the determinant, interpreting them as cycles in a graph. This yields an elementary and intuitive argument, which in the finite case recovers a version of Loewy's classical matrix result from [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23--64].