An arXiv preprint dated 28 August 2026 reports an algebraic proof that a family of difference-power matrices has a nonzero determinant throughout the conjecture's stated threshold range. The theorem applies when the matrix dimension is even, the real nodes are pairwise distinct, and the integer exponent is at least one less than the dimension.
That result is presented as a completion of Colombo's determinant conjecture. Its scope is precise: the argument covers the specified even dimensions and real, distinct-node setting, not cases outside those hypotheses.
The question behind the matrix
The conjecture asks whether the determinant of this difference-power matrix remains nonzero for every integer exponent at or above the threshold. In plain terms, it asks whether the matrix stays nonsingular across the entire allowed range, rather than failing at some particular exponent.
The matrix dimension, denoted n in the paper, must be even and at least 2. Its inputs are real coordinates with no repeats. The exponent, denoted d, is an integer, and the threshold is one less than n.
A contradiction built from factors
The proof's central move is to assume the matrix is singular and choose a nonzero kernel vector, meaning coefficients that would make the matrix product vanish. From those coefficients and the node-associated pure powers, it constructs a binary form, a homogeneous polynomial in two variables.
An apolar pairing and pure-power evaluation then connect the matrix entries with evaluations of that form. This lets the authors turn the presumed singularity into a statement about the form's real factors.
At the heart of the odd-exponent argument is a count of factors. The constructed form has at least n + 1 real projective linear-factor occurrences, counted with multiplicity, while its real Waring length, the number of real powers needed in its representation, is no more than n. The paper records the contradiction as . Here, n is the number of nodes, F is the nonzero real binary form, tau(F) counts its real projective linear factors with multiplicity, and LR(F) is its real Waring length. The left side cannot be reconciled with the right side, so the assumed singularity cannot occur.
A separate Vandermonde-independence step, which checks that the relevant node-based powers remain independent, ensures that F is not the zero form when the number of nodes is no greater than one plus the exponent. That matters because the factor count is being applied to a genuine, nonzero form.
How the proof closes the range
The full completion uses more than the odd case. It combines Colombo's threshold result, the odd-exponent proof and cited distance-power results for even exponents. Together, those ingredients cover every integer exponent at or above one less than the matrix dimension.
Under the theorem's hypotheses, the resulting conclusion is that the determinant is nonzero throughout that range. Because this is a deductive proof about mathematical objects, the claim is conditional on the stated assumptions and invoked results, rather than an estimate based on observations.
The result goes beyond nonsingularity
The paper also gives an exact rank formula for all nonnegative integer exponents under the same node and dimension assumptions. The rank is the smaller of the matrix dimension and the exponent plus one. In other words, the formula sets a ceiling at n while allowing rank to track d + 1 below that ceiling.
It also characterizes the determinant's sign for exponents at or above the threshold. The determinant is positive when d is odd; when d is even, its sign follows the rule given by negative one raised to half the matrix dimension.
The odd-exponent derivation is reported as kernel-checked in Lean 4.32.1 with mathlib v4.32.1, a formal proof environment and supporting library. The paper says the formalization contains no project-specific axiom or sorry/admit placeholder. It also explicitly says kernel acceptance does not replace external peer review.
The manuscript states that the formalization source, code and detailed documentation are hosted at the GitHub repository named in the paper.
The boundaries are part of the result
The boundary conditions matter. The supplied analysis does not establish analogous statements for odd matrix dimensions, repeated real nodes or noninteger exponents. The even-exponent part relies on cited distance-power results rather than full proofs reproduced in this manuscript.
The formal verification described in the preprint concerns the odd-exponent derivation; the text does not say that the entire even branch has been formally verified. The authors frame the kernel check as a verification aid, not as a substitute for peer review or a way to settle literature priority.
The arXiv record identifies this as version 1 dated 28 August 2026. The paper studies a mathematical family rather than a sampled population, so its conclusion is a deductive statement under stated assumptions, not an estimate from observations.
Paper data and sources
Original title: An algebraic proof of Colombo's difference-power determinant conjecture
Authors: Kun Li, Li Tie, Peng Wang, Zihan Liu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text