Iterate Wronskians over $\mathbb{R}^d$ as $N$-ary brackets on $\mathbb{R}[x^1,\ldots,x^d]$: the $N$-bonacci numbers bound the highest total degrees

Arthemy V. Kiselev; Markuss G. Kenins

REVIEW 3 major objections 3 minor 10 references

Iterate Wronskians over $\mathbb{R}^d$ as $N$-ary brackets on $\mathbb{R}[x^1,\ldots,x^d]$: the $N$-bonacci numbers bound the highest total degrees

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For any starting polynomials, iterating the generalized Wronskian grows the highest degree no faster than the N-bonacci numbers.

desk verdict Main theorem is false: span-closure allows combining any two accumulated monomials, giving 2^n degree growth for d=1,k=1,N=2, not Fibonacci. read the letter →

arxiv 2607.26039 v1 pith:WRNUWRSI submitted 2026-07-28 math.RA math-phmath.COmath.MPmath.NTmath.QA

Markuss G. Kenins , Arthemy V. Kiselev This is my paper · ORCID

classification math.RAmath-phmath.COmath.MPmath.NTmath.QA MSC 05E1511B3915A1505A1005A1617B65

keywords DifferentialpolynomialN-aryLiebracketmultivariateWronskiandeterminantN-bonaccinumbersasymptoticdegreegrowthmonomialrecurrencedimensionzero-detectioninlinearrecurrences

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper considers an iterative process on polynomials in d variables: take N polynomials, compute their complete generalized Wronskian determinant (a determinant built from partial derivatives up to order k), treat the result as a new polynomial, and keep re-using the newest polynomials as inputs. The question is how fast the highest total degree can grow. The main theorem gives a universal ceiling: after n iterations, the highest total degree—and the highest degree in each separate variable—grows asymptotically no faster than the nth N-bonacci number, the Fibonacci-like sequence in which each term is the sum of the previous N terms. This matters because degree growth bounds how much new polynomial space the iteration can span, connecting the problem to a classical question about growth of Lie algebras generated by iterated brackets. The ceiling is exact for one-variable polynomials with odd-order Wronskians, and the authors conjecture—with extensive computer evidence—that it is exact in every dimension and differential order.

What carries the argument

The engine of the proof is the complete generalized Wronskian determinant W^k_d of differential order k over R^d, viewed as an N-ary bracket with N = binom(d+k,d). For monomial inputs, the output is again a monomial; its degree in each variable is the sum of the input degrees in that variable minus Nk/(d+1), and its coefficient is a determinant built from the exponents. This turns degree evolution into a linear recurrence with constant coefficients. The corresponding N-bonacci recurrence has characteristic polynomial λ^N − λ^{N−1} − ⋯ − λ − 1, whose unique root α_N above 1 lies between the golden ratio and 2, while all other roots lie inside the unit disk. That separation of roots converts t

What would settle it

Take d=1, k=1, N=2, start with x^2 and x^3, and at each step compute the Wronskian of every pair among the accumulated polynomials. Record the maximum total degree. If the maximum degree ever exceeds the nth Fibonacci number or shows an asymptotic exponent greater than φ≈1.618, the theorem's bound collapses; if it stays below the bound, the single-chain reduction is supported.

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Extended reading notes

Core claim

The central claim is that iterating the complete generalized Wronskian W^k_d as an N-ary bracket on R[x^1,...,x^d], where N = binom(d+k,d), cannot make degrees explode faster than the N-bonacci numbers. For any initial N-tuple of polynomials, the sequence of highest total degrees d^{(N)}_n satisfies lim_{n→∞} d^{(N)}_n / F^{(N)}_n < ∞, and the same holds for the highest degree in each coordinate. The argument reduces to monomial seeds: the Wronskian of monomials is again a monomial, and its degree in each variable is the sum of the input degrees minus the constant Nk/(d+1). Along the fastest single chain of re-use, this yields a linear recurrence that, after homogenization, is exactly the N-

Load-bearing premise

The bound rests on the assumption that the fastest growth comes from a single window of length N—using the newest output immediately and dropping the oldest argument—so that degrees obey one N-term recurrence; if choosing arbitrary N-tuples from the whole current span can create branched, faster growth, that recurrence argument would not cover the stated problem.

Editorial extensions

If this is right

For any starting polynomials, after n iterations the highest total degree is eventually no larger than the nth N-bonacci number; the same ceiling applies to the highest degree in each variable.
The dimension of the generated polynomial subspaces grows at most like (α_N^d)^n, which is strictly below the naive (2^d)^n bound.
In one variable with odd differential order k, the ceiling is attained: there exist monomial seeds whose degrees are the N-bonacci numbers up to a shift and whose coefficients never vanish, so the bound is sharp there.
If the paper's conjecture is correct, the N-bonacci ceiling is asymptotically sharp in every dimension and differential order, not just on the line.
Whether a particular seed choice produces nonzero coefficients forever is a zero-detection problem for an integer linear recurrence; the paper reports many cases where coefficients stay nonzero for about 10,000 iterations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof as written bounds a single chain of reuse (newest output always enters immediately, oldest drops out). If the problem is read literally as allowing any N polynomials from the whole current span at each step, branching could in principle accelerate growth; a direct small-scale test would show whether the ceiling still holds under that wider reading.
The constant term Nk/(d+1) in the per-variable degree recurrence suggests the total-degree bound may be looser in higher dimensions; tracking exact constants could yield a sharper effective bound for specific seeds.
Because coefficient nonvanishing is a zero-detection problem for linear recurrences, the conjecture implies that certain constructed integer sequences never hit zero; proving that for even one nontrivial seed would be a concrete positive case of a known hard decision problem.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The central theorem is false under the paper's own definition. The research problem closes A_{n+1} = span(A_n ∪ all Wronskians of N elements of A_n), but Appendix A proves a bound for a different process: a single sliding window where each new output is immediately re-used and the oldest argument is dropped. That sketch misses that you can combine any two large elements already present. For d=1, k=1, N=2, start with x^2 and x^3. By induction A_n contains every monomial of degree from 2 to 2^n + 2: the two largest give W(x^{d_n}, x^{d_n-1}) = x^{2d_n-2}, so d_{n+1} = 2d_n - 2 and hence d_n = 2^n + 2. Fibonacci numbers grow like φ^n, so d_n/F_n → ∞, contradicting both the theorem and the dimension corollary.

That said, the paper is not empty. Examples 1–3 correctly exhibit N-bonacci degree growth for the restricted sliding-window iteration scheme; that is a useful observation. Appendix B re-proves the known growth rate of N-bonacci numbers with Rouché's theorem and a geometric bound; the comparison with Wolfram's constant is a small but clean contribution. The computer experiments and the link to the Skolem–Pisot problem are also worth a look.

The soft spot is exactly the gap between the stated span-closure problem and the proof. The proof's assertion that not using the newest argument immediately cannot be faster is simply wrong: in the counterexample, at n=3 the fastest output comes from pairing two older monomials, not from the most recent one. This is not a minor fix; the entire upper-bound claim depends on restricting to the sliding-window scheme. If the paper were reframed as a theorem about that scheme, the examples would support it, but the universal bound for the span closure is refuted.

Who is this for? Readers interested in Wronskian N-ary brackets and Kirillov's problem might want to see the examples, but they should not trust the main theorem. It deserves a serious referee—not because the result is plausible, but because the error is subtle enough to slip past and the examples are genuinely interesting. I would recommend peer review with a request for major revision: either repair the proof by addressing branching, or honestly restrict the claim to single-chain iteration.

Referee Report

3 major / 3 minor

Summary. The paper studies iteration of the complete generalized Wronskian determinant W_k^d on the polynomial algebra R[x^1,...,x^d], viewed as an N-ary bracket with N=binom(d+k,d). The formal research problem defines A_{n+1}=span(A_n ∪ B_{n+1}), where B_{n+1} contains the Wronskians of arbitrary N-tuples of elements of A_n, and d_n is the highest total degree appearing in A_n. The main theorem claims that for any initial arguments, d_n grows asymptotically no faster than the N-bonacci numbers, and a corollary bounds dim(A_n) by (α_N^d)^n < (2^d)^n. The paper also gives examples attaining the bound for d=1 and odd k, discusses the coefficient nonvanishing problem via Skolem–Pisot, and conjectures attainment in general.

Significance. If the theorem were correct, the paper would provide a general growth bound for a natural polynomial analogue of Kirillov's problem and a dimension bound for subspaces generated by iterated Wronskians. The paper contains correct elementary material on N-bonacci characteristic roots and explicit, reproducible monomial examples. However, the central theorem is false under the paper's own span-closure definition, and the dimension corollary fails with it. The counterexample is simple, within the stated research problem, and does not depend on subtle interpretations. The paper's contribution therefore reduces to a set of examples and a proof for a restricted sliding-window iteration scheme, not the theorem advertised in the abstract and conclusion.

major comments (3)

[Research problem and Theorem, p.5; Appendix A] The theorem is false as stated. Take d=1,k=1,N=2 and A0=span{x^2,x^3}. Since W(x^a,x^b)=(b-a)x^{a+b-1}, induction gives A_n=span{x^2,...,x^{2^n+2}}: pairing the two largest monomials x^{2^n+1} and x^{2^n+2} produces x^{2^{n+1}+2}, and no pair gives a larger exponent. Hence d_n=2^n+2. The 2-bonacci numbers F_n grow like φ^n, so d_n/F_n→∞, contradicting the Theorem's assertion lim d_n/F_n<∞. This is a direct counterexample to the central claim.
[Corollary, Eq. (5), p.5] For the same initial data, A_n contains exactly one monomial of each degree from 2 to 2^n+2, so dim(A_n)=2^n+1. The corollary claims dim(A_n)≲(α_2^d)^n=φ^n, which is impossible because (2^n+1)/φ^n→∞. Thus the dimension bound fails together with the theorem.
[Appendix A] The proof reduces d_n to a single N-window recurrence a_n=a_{n-1}+...+a_{n-N}-kN/(d+1) by asserting that fastest growth requires immediate re-use of the newest monomial and dropping the smallest-degree old one. This assertion is not justified by the research problem's definition A_{n+1}=span(A_n∪B_{n+1}) with B_{n+1} consisting of Wronskians of arbitrary N elements of A_n. The counterexample above uses exactly this freedom by pairing the two largest elements at each step. The Appendix therefore proves at most an upper bound for the sliding-window scheme m_n:=W(m_{n-N},...,m_{n-1}) used in Examples 3–4 and the Conjecture, not for the stated theorem.

minor comments (3)

[Title and running head] The title contains OCR-type typos ('ITERA TE', 'OVERR'); please proofread the manuscript carefully.
[Appendix A and Example proofs] The paper repeatedly invokes [2, Theorem 7] for the structure of Wronskians of monomials and for generalized Vandermonde determinants, but [2] is an unpublished preprint and the formula is not stated. A self-contained statement of this lemma would improve readability and verifiability.
[Example 4] The coefficient recurrence c_n=-c_{n-1}-c_{n-2}+c_{n-3} is asserted without derivation, and the statement that coefficients are nonzero 'up to n≈10,000' is experimental evidence, not a proof of nonvanishing. The distinction should be made explicit in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified: the main theorem's reliance on [2, Thm 7] is a standard parameter-free lemma, and the contested sliding-window reduction in Appendix A is a proof gap rather than an equation-level circularity.

full rationale

After walking the derivation chain, I find no step where the paper's conclusion is equivalent to its inputs by construction, nor a fitted parameter renamed as a prediction, nor a load-bearing self-citation. The proof of the main theorem invokes [2, Thm 7] for the formula that a Wronskian of monomials is a monomial whose degree in each variable is the sum of the input degrees minus kN/(d+1). This is a parameter-free determinant lemma with stated assumptions that do not include the N-bonacci bound, so its use is independent evidence rather than circularity. The only load-bearing move in Appendix A is the sentence "If the newly-created argument was not used immediately in the next iteration, the growth was necessarily not faster than maximal" and the resulting assertion that a_n satisfies a_n = a_{n-1}+...+a_{n-N}-kN/(d+1). That step reduces the general span-closure process to a single sliding-window recurrence; it is not derived from the paper's own definition A_n = span(A_{n-1} ∪ B_n). This is an unsupported (and, for d=1,k=1,N=2 with initial x^2,x^3, apparently false) mathematical claim, but it is a correctness gap rather than a circular definition or fitted prediction. Examples 1-3 are explicit constructions, and Appendix B independently reproves the N-bonacci root bound. Accordingly no circular step is exhibited, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the authors' prior theorem [2] for the monomial Wronskian formula and on the span-closure definition in the research problem. No free parameters are fitted. The invented-entities ledger is empty.

assumptions (4)

domain assumption Wronskian of monomials formula [2, Thm 7]: the complete generalised Wronskian of N monomials is a single monomial with exponent vector equal to the sum of input exponent vectors minus kN/(d+1) in each variable, times a generalized Vandermonde constant.
Invoked in Appendix A and Example 4; it is a theorem from the authors' own earlier preprint [2], not proved here.
domain assumption Span-closure definition of A_n: A_{n+1}=span(A_n ∪ all Wronskians of N-tuples from A_n), and d_n is the maximum total degree in A_n.
Defined in the Research problem on p.4 and used for the Theorem and Corollary. This is the definition that makes the counterexample valid; the paper's examples use a more restrictive sliding-window scheme.
standard math Standard theory of linear recurrences: closed-form solution a_n = a^* + C α_N^n + Σ c_j λ_j^n with |λ_j|<1.
Used in the proof of the main theorem and the Proposition. These are standard results in the theory of linear recurrences.
standard math Rouché's theorem from complex analysis.
Used in Appendix B to locate the roots of the N-bonacci characteristic polynomial.

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Cite this review

Pith. "Pith review of Iterate Wronskians over $\mathbb{R}^d$ as $N$-ary brackets on $\mathbb{R}[x^1,\ldots,x^d]$: the $N$-bonacci numbers bound the highest total degrees." pith.science (2026). https://pith.science/paper/WRNUWRSI

@misc{pith2026260726039,
  author       = {Pith},
  title        = {Pith review of: Iterate Wronskians over $\mathbbR^d$ as $N$-ary brackets on $\mathbbR[x^1,\ldots,x^d]$: the $N$-bonacci numbers bound the highest total degrees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WRNUWRSI}},
  note         = {Machine review of arXiv:2607.26039}
}

abstract

For the algebra $\mathbb{R}[x^1,\ldots,x^d]$ of polynomials in $d\geqslant 1$ variables, regard the complete generalised Wronskian $W_d^k$ of differential order $k\geqslant 1$ over $\mathbb{R}^d$ as the $N=\tbinom{d+k}{d}$-ary Lie bracket. Take an $N$-tuple of polynomials, calculate their Wronskian, and keep re-using the newly-created polynomials to produce more of them. The problem is: how fast do their maximal total degrees grow with the number $n$ of iterations of the bracket? Here enter the $N$-bonacci numbers defined by the recurrence $F^{(N)}_n=F^{(N)}_{n-1}+\cdots+F^{(N)}_{n-N}\in \mathbb{N}$. We prove that for any choice of the initial arguments, the sequence of highest total degrees $d^{(N)}_n \geqslant 0$ grows (if at all) asymptotically no faster than the $n$th $N$-bonacci number: $\lim_{n\to+\infty} (d^{(N)}_n/F^{(N)}_n )<\infty$. We show that for $d=1$ and $k$ odd, the highest polynomial degrees do attain the $N$-bonacci bound.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 3 linked inside Pith

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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D., Portier N

Blondel V . D., Portier N. (2002) The presence of a zero in an integer linear recurrent sequence is NP-hard to decide.Linear Algebra Its Appl.351–352, 91–98

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arXiv 2026

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A., Kontsevich M

Kirillov A. A., Kontsevich M. L. (1983) The growth of the Lie algebra generated by two generic vector fields on the line.Vestnik Moskov. Univ. Ser. I Mat. Mekh., no. 4, 15–20

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