REVIEW 3 minor 8 references
Powers of matrices with all principal minors equal to 1
T0 review · 0 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read The paper proves that a matrix over any quotient of a commutative ring by an integrally closed ideal — including every reduced ring and every ring Z/d — has all its powers (positive and negative) sharing the property that all principal mino
desk verdict A solid, genuinely new generalization of Putnam 2021 B5 with a delicate but sound combinatorial core; the AI provenance disclosure is the only real concern, and it argues for independent verification rather than distrust. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Two components carry the argument. The universal integrality theorem (Theorem 4.6) uses Viète's formulas and a determinant expansion: for each size r, the elementary symmetric functions of the weights of Hamilton cycles on an r-element set lie in the ideal generated by shorter cycle weights and principal-minor defects, forcing each cycle weight to be integral over J(A). To establish the needed ideal-membership bounds, the paper proves a purely combinatorial lemma (Lemma 4.3): the multiset union of two distinct Hamilton cycles on a finite set can be partitioned into at least three cycles, each having length strictly smaller than the set's size. This lemma is what lets the author place product
What would settle it
Enumerate all pairs of distinct Hamilton cycles on a finite set (e.g., size 5) and check whether their multiset union always admits a partition into at least three cycles of length < |S|; one counterexample would falsify Lemma 4.3 and therefore the main theorem. On the ring side, a finite-ring search (e.g., over Z/4) for a 1-principled matrix whose square is not 1-principled would directly contradict Corollary 5.4 and the main theorem.
Extended reading notes
Core claim
The paper proves Theorem 1.2: if I is an integrally closed ideal of a commutative ring D and A is a matrix over D/I whose principal minors are all 1, then every power A^m (m any integer) also has all principal minors 1. The key upstream result is Theorem 4.6: for any n×n matrix A over any commutative ring, the weight of every nontrivial cycle — a product of entries along a cycle — lies in the integral closure of the ideal J(A) generated by the differences between principal minors and the corresponding products of diagonal entries. When J(A) lies in an integrally closed ideal I, reducing modulo I forces all cycle weights to vanish, so A is nullcyclic; nullcyclicity is inherited by powers via
Load-bearing premise
The argument's load-bearing premise is Lemma 4.3, the combinatorial assertion that the multiset union of two distinct Hamilton cycles on a finite set can be partitioned into at least three cycles each shorter than the set; the main theorem follows only if this decomposition always exists.
Editorial extensions
If this is right
- Over every reduced ring, a 1-principled matrix has all integer powers 1-principled (Corollary 4.9).
- Over Z/d for any integer d, including the original Putnam setting d=2, every power of a 1-principled matrix is 1-principled (Corollary 5.4).
- The same conclusion holds over quotients of Prüfer domains by arbitrary ideals and over principal quotients of normal domains (Corollaries 5.8 and 5.10).
- Without the 1's: a principled matrix over such rings has all nonnegative powers principled (Theorem 1.9).
- The universal integrality theorem gives a new ideal-theoretic constraint: even when cycle weights do not vanish, they are always integral over J(A), so their behavior is controlled by integral closure rather than by ad hoc computations.
Reading between the lines
- The main theorem suggests classifying rings over which the Putnam property holds in full; the paper's evidence points to quotients by integrally closed ideals, and a natural testable question is whether every ring with the property admits such a presentation.
- Lemma 4.3 may be useful beyond matrices: any weight function on arcs of a complete digraph that vanishes on short cycles forces products over Hamilton cycles to lie in powers of the ideal generated by short-cycle weights, suggesting a general graph-theoretic integrality principle.
- The walk-expansion proof of nullcyclicity inheritance works over any ring, so 1-nullcyclicity is a genuinely ring-independent sufficient condition; one could explore a 'nullcyclic up to nilpotents' variant that might extend the result beyond integrally closed quotients.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies matrices over commutative rings whose principal minors are all 1 (called 1-principled) and asks whether this property is preserved under taking powers. The main theorem (Theorem 1.2) proves it for quotients D/I where I is integrally closed, yielding the cases Z/d, quotients of Prüfer domains, principal quotients of normal domains, and all reduced rings (Corollaries 4.9, 5.4, 5.8, 5.10). The proof introduces stronger conditions — principled, nullcyclic, 1-nullcyclic — proves they are stable under powers (Propositions 2.3, 3.5 and Lemma 3.4), and then bridges from principled to nullcyclic through a universal integrality theorem (Theorem 4.6): every nontrivial cycle weight is integral over the ideal generated by principal-minor defects. The engine of Theorem 4.6 is a combinatorial lemma (Lemma 4.3) on partitioning the multiset union of two distinct Hamilton cycles into short cycles, together with an induction using elementary symmetric polynomials and Viète's formulas.
Significance. If the results hold, this is a clean and substantive generalization of Putnam 2021 B5, moving from Z/2 to a broad class of rings. The paper identifies 1-nullcyclicity as the ring-independent property that explains stability under powers, and the universal integrality statement of Theorem 4.6 is a valuable result in its own right. The exposition is detailed, largely self-contained, and gives explicit examples and a counterexample over a non-reduced ring. I examined the most delicate input, Lemma 4.3; its proof is sound (for |S|=2 the hypothesis is vacuous because the two Hamilton cycles coincide up to rotation). The walk-sum arguments and the Vieta induction in Theorem 4.6 are coherent. I found no load-bearing mathematical error.
minor comments (3)
- [§4, Theorem 4.6] After equation (4.4), the text invokes Lemma 4.2(d) and concludes "L = \overline{J(A)}". The correct conclusion is "\overline{L} = \overline{J(A)}", or one can directly use the already-established inclusion L ⊆ \overline{J(A)} to get \overline{L} ⊆ \overline{J(A)}. As printed, the equality is generally false and the cited lemma concerns integral closures. This is a typographical slip and does not affect the proof's conclusion.
- [Footnote 6] The statement that the 4×4 counterexample is "minimal in the sense that there are no matrices of size ≤ 3 that are 1-principled but have non-1-principled powers" is asserted without proof or reference. This is not load-bearing, but it should be justified or explicitly marked as an empirical observation.
- [§1–§4] The notation "\overline{I}" for integral closure is introduced in Definition 4.1, and subsequently the same overline is used for ideals such as \overline{L} in Theorem 4.6. The proof would be easier to read if the overline notation were also explained in a sentence just before its first use in Lemma 4.2, or if the authors consistently say "integral closure" at the first few occurrences.
Circularity Check
No significant circularity: the central derivation is self-contained and the self-citations are non-load-bearing.
full rationale
The paper's main theorem (Theorem 1.2) is derived from Theorem 5.2 and Corollary 3.6, both proved in the paper from definitions and earlier lemmas. The key upstream result is Theorem 4.6, which proves that every nontrivial cycle's A-weight is integral over the principal-minor-defect ideal J(A); its proof uses an induction on the cycle length r, the determinant expansion (2.1), Lemma 4.3 (the combinatorial partition of two Hamilton cycles into at least three shorter cycles), Lemma 4.5, and Viète's formulas. None of these steps assumes the conclusion it is used to prove. The induction in Theorem 4.6 is a genuine strong induction: the shorter-cycle weights are covered by the induction hypothesis, and the Hamilton-cycle weights are handled by the symmetric-polynomial argument. Lemma 4.3 is proved directly by constructing the cycles C and C' from a vertex whose outgoing arcs differ and then partitioning the remaining balanced multidigraph. Proposition 2.3 and Lemma 3.4 prove the nullcyclic and 1-nullcyclic stability properties from the walk expansion (2.2), again without circularity. The only references to the author's prior work are [2], used for background, a counterexample, and an optional alternative route in Remark 3.3; the main proof does not rely on [2] for any load-bearing step. The other references are standard facts about integral closure and Prüfer domains. There is an apparent typo in Theorem 4.6's final application of Lemma 4.2(d) ('L = J(A)' instead of the intended 'overline{L} = overline{J(A)}'), but the text immediately uses w_A(H) ∈ overline{L}, so the intended argument is unchanged. This is a minor presentational defect, not circularity.
Assumptions & free parameters
assumptions (5)
- domain assumption R is a commutative, associative, unital ring.
- standard math The walk expansion formula (A^m)_{i,j} = sum of A-weights of length-m walks is valid.
- standard math A balanced directed multigraph's arcs can be partitioned into directed cycles.
- standard math Properties of integral closure of ideals (Lemma 4.2) hold as stated, including that the integral closure of an ideal is an integrally closed ideal.
- domain assumption In a quotient ring D/I with I integrally closed, an element integral over I lies in I.
invented entities (3)
-
1-principled matrices
-
principled matrices
-
1-nullcyclic / nullcyclic matrices
Cite this review
Pith. "Pith review of Powers of matrices with all principal minors equal to 1." pith.science (2026). https://pith.science/paper/EULD6KVY
@misc{pith2026260628976,
author = {Pith},
title = {Pith review of: Powers of matrices with all principal minors equal to 1},
year = {2026},
howpublished = {\url{https://pith.science/paper/EULD6KVY}},
note = {Machine review of arXiv:2606.28976}
}
abstract
We say that a square matrix $A$ is \emph{$1$-principled} if all its principal minors are equal to $1$. We show that over any well-behaved ring, any power $A^m$ of a $1$-principled matrix $A$ is again $1$-principled. Well-behaved rings include all reduced rings as well as all quotients of commutative rings modulo integrally closed ideals; in particular, all fields and all quotients of $\mathbb{Z}$ are well-behaved. We note that $m$ can be any integer, positive or negative. This generalizes Problem B5 of the 2021 Putnam contest in multiple directions. Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrix $A = \left(a_{i,j}\right)_{i,j\in\left[n\right]}$ is \emph{$1$-nullcyclic} if all its diagonal entries are $1$ and if all the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish. We show that if $A$ is $1$-nullcyclic, then so is $A^m$ for any integer $m$. Furthermore, every $1$-nullcyclic matrix is $1$-principled over any commutative ring, while the converse holds if the ring is well-behaved. Along the way, we prove analogous results that don't require the diagonal entries to be $1$. These are concerned with \emph{principled matrices} (those whose principal minors equal the respective products of diagonal entries) and \emph{nullcyclic matrices} (those whose cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish); their diagonal entries can be arbitrary. A crucial auxiliary result, which holds for any $n\times n$-matrix $A$, is that the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ (with $k>1$ and distinct $i_1,i_2,\ldots,i_k$) are integral over the ideal generated by the principal minors of $A$ minus the corresponding products of diagonal entries of $A$.
Reference graph
Works this paper leans on
-
[2]
Darij Grinberg,On the principal minors of the powers of a matrix, updated version of a paper published in Gazeta Matematica (2022, no. 1–2, pp. 1–13), arXiv:2204.07885v2. https://arxiv.org/abs/2204.07885v2
arXiv 2022
-
[6]
336, Cambridge University Press, Cambridge, 2006
Irena Swanson and Craig Huneke,Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. https://www.math.purdue.edu/~iswanso/book/SwansonHuneke.pdf
2006
-
[1]
Ullman and Paul Zeitz,The Eighty-Second William Lowell Putnam Math- ematical Competition, Amer
Daniel H. Ullman and Paul Zeitz,The Eighty-Second William Lowell Putnam Math- ematical Competition, Amer. Math. Monthly129(2022), no. 8, 703–716. doi:10.1080/00029890.2022.2099714
arXiv 2022
-
[3]
Huckaba and Ira J
Marco Fontana, James A. Huckaba and Ira J. Papick,Prüfer Domains, Monographs and Textbooks in Pure and Applied Mathematics, vol. 203, Marcel Dekker, New York, 1997
1997
-
[4]
https://www.cip.ifi.lmu.de/~grinberg/t/22s/graphs.pdf
Darij Grinberg,An introduction to graph theory, 19 May 2026. https://www.cip.ifi.lmu.de/~grinberg/t/22s/graphs.pdf
2026
-
[5]
Henri Lombardi, Claude Quitté,Commutative algebra: Constructive methods. Fi- nite projective modules, updated version of a book published by Springer in 2015, arXiv:1605.04832v4
arXiv 2015
-
[7]
R. P. Stanley,Enumerative Combinatorics, Volume 1, second edition, Cambridge Studies in Advanced Mathematics49, Cambridge University Press, 2012
2012
-
[8]
Doron Zeilberger,A combinatorial approach to matrix algebra, Discrete Mathe- matics56(1985), pp. 61–72. Seehttps://sites.math.rutgers.edu/~zeilberg/ mamarimY/DM85dg.pdffor an updated version. 20
1985
Reviewed August 4, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.