Determinantally Equivalent Functions Beyond the Nowhere-Zero Case

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].

• The paper establishes that under property D, determinantally equivalent functions with zeros are canonically related through conjugation or transposition.
• It employs an elementary combinatorial proof based on cycle structure analysis to bypass traditional nowhere-zero requirements.
• The findings extend principal minor theory, providing precise structural conditions for kernel equivalence in diverse nondegenerate settings.

Determinantally Equivalent Functions and Transformations in the Presence of Zeros

Introduction and Context

The investigation targets the structural characterization of pairs of functions $K, Q : \Lambda^2 \to \mathbb{F}$ (with $\Lambda$ arbitrary and $\mathbb{F}$ a field) that are determinantally equivalent; that is, for every finite collection of points $x_1, \ldots, x_n \in \Lambda$, the determinants of the associated matrices built from $K$ and $Q$ coincide, i.e.,

$$\det(K(x_i,x_j))_{i,j=1}^n = \det(Q(x_i,x_j))_{i,j=1}^n, \text{ for all } n \geq 1.$$

Prior work established that, in the symmetric case, such determinantally equivalent kernels are related by canonical "conjugation" (multiplication by a nowhere-zero function) and/or transposition. However, without symmetry, the conclusion can fail unless additional structural properties are imposed. The key structural condition is class $\mathcal{D}$: a nondegeneracy requirement on the $2\times2$ minors of the function values, which generalizes, in the finite case, conditions known from principal minor theory.

Previous results required a "nowhere-zero" assumption on $K$ and $\Lambda$0 away from the diagonal. This paper advances the theory by completely removing the nowhere-zero hypothesis, thus significantly expanding the class of functions covered. The main technical achievement is a complete and elementary combinatorial proof, based on cycle structure analysis, of the fundamental equivalence—eschewing all references to linear algebra or spectral techniques.

Main Results

The central theorem can be summarized as follows:

If $\Lambda$1 and $\Lambda$2 are determinantally equivalent, both are of class $\Lambda$3, then $\Lambda$4 can be obtained from $\Lambda$5 through some combination of conjugation by a (possibly vanishing) function and/or transposition, even when $\Lambda$6 and $\Lambda$7 admit zeros throughout their domain.

The explicit content is that for every such pair, either

• $\Lambda$8 for some (possibly zero-valued) function $\Lambda$9, or
• $\mathbb{F}$0

where the transformations correspond, respectively, to conjugation and conjugation of the transpose.

The proof establishes that the sole obstructions to this conclusion are precisely those ruled out by property $\mathbb{F}$1—all "pathological" zero patterns that prevent unification via the canonical transformations are combinatorial in nature and do not survive the structural constraint.

Combinatorial Approach and Technical Framework

The previous analysis in [Mantelos (2026)] utilized the permutation-based expansion of determinants, identifying $\mathbb{F}$2-cycles and $\mathbb{F}$3-cycles as the locus of potential structural failure. The present work constructs an entirely combinatorial mechanism for tracking the influence of zero values in $\mathbb{F}$4 and $\mathbb{F}$5 on these cycles via an augmented graphical framework. Key technical instruments are:

• Classification of cycles: Each $\mathbb{F}$6-cycle $\mathbb{F}$7 is shown to fall into exactly one of two structural cases (analogous to left- and right-action compatibility by conjugation or transposition).
• Propagation and propagation obstruction: The structure of zeros is shown to be extremely rigid under $\mathbb{F}$8; the only compatible patterns make global transformations possible.
• Consistency identities: The paper develops new identities to extend those previously obtained under the nowhere-zero assumption, now accommodating vanishing terms. Essential use is made of the graphical language to guarantee consistency of transformation definitions across overlapping cycles.

There is an intricate analysis, with case breakdowns based on the occurrence and arrangement of zeros in the function, and their implications for the possible cases to which a cycle can belong. These ensure the cocycle properties required for the global transformation to exist.

Implications

Theoretical Impact

• Extension of Principal Minor Theory: This result gives the sharpest possible structural theorem in the functional analog of the principal minor problem (e.g., see Loewy (1986)), resolving issues left open when zeros appear.
• Fundamental rigidity: Property $\mathbb{F}$9 is shown to be the precise threshold for rigidity: no structurally distinguishable configurations exist beyond what is classified by conjugation and transposition.
• Transfer to Infinite Domains: The techniques adapt not only to the finite matrix case (recovering and refining classical results), but to abstract kernel settings, relevant for operator theory, integral equations, and probabilistic processes in the continuum.

Practical Implications

In the context of determinantal point processes (DPPs) and related random matrix theory, this work sharpens the uniqueness and identifiability analysis of kernels: all nontrivial equivalences of correlation kernels are already captured by the known symmetries, subject to weak regularity (i.e., class $x_1, \ldots, x_n \in \Lambda$0), with no further equivalences introduced by possible zeros in the kernel.

It also points toward robust algorithmic strategies for identifying canonical representatives of equivalence classes, even in highly degenerate or structured-zero settings.

Future Directions

• Strengthening of Class $x_1, \ldots, x_n \in \Lambda$1 Theory: The precise extremal structure of class $x_1, \ldots, x_n \in \Lambda$2 functions deserves further investigation, especially in infinite or measurable settings.
• Relaxation and Measure Theory: Integration of measure-theoretic constraints or treatment of function spaces (Banach or Hilbert kernels) could generalize the result beyond pointwise (algebraic) structure.
• Connection to Inverse Problems: Applications to identifiability and uniqueness in inverse spectral and stochastic problems are an open avenue, especially where only partial minors/determinants can be observed.

Conclusion

This paper conclusively establishes that, under property $x_1, \ldots, x_n \in \Lambda$3, the full set of determinantally equivalent functions (even with zeros present) are explicitly classified by canonical conjugation and transposition actions. The combinatorial-graphical machinery introduced not only subsumes prior nowhere-zero cases but does so without recourse to linear algebra, indicating a deep combinatorial foundation for determinant-based equivalences (2604.03934). The result both clarifies the functional version of the principal minor problem and strongly informs the uniqueness theory for determinantal processes and kernel-based models.