REVIEW 3 major objections 3 minor 12 references
Periodicities in the Riordan arrays of polynomials over finite fields
T0 review · 3 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash
Pith's one-line read Over a finite field, every column of a polynomial Riordan array is eventually periodic, and the periodic blocks are powers of one circulant matrix applied to the zeroth column.
desk verdict New and mostly correct transfer-matrix results for Riordan arrays over finite fields, but the proof of Lemma 1 misses the degree case the main theorem needs; still worth refereeing. 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
The driving object is the circulant matrix generated by the coefficient vector of the polynomial in the second argument: A = circ(c_{d3},...,c_0) padded to size π, with coefficients folded modulo π when π < d3+1. Because multiplication by t p3 shifts and convolves coefficients, the periodic block of the next column is the product of A with the current block; induction on k gives C_k = A^k C_0. The period π itself comes from the standard fact that coefficients of a rational function over a finite field form a linear recurring sequence whose least period is the order of its minimal polynomial, which the paper shows is the reciprocal of p2 up to a scalar.
What would settle it
Compute column 1 of the Riordan array ((1+t)/(1+t+t^2), t(1+t)) over F_2 directly: its periodic block, extracted from the coefficient table, must equal circ(1,1,0) times the zeroth-column block. If the product differs from the directly extracted block, Theorem 4(ii) fails; if it matches, the identity survives the simplest possible check.
Extended reading notes
Core claim
On its own terms, the central claim is Theorem 4: under the hypotheses p_i(0)≠0, p2 coprime to p1 and p3, and d2 ≤ d1+1, the k-th column (k≥1) of the Riordan array (p1/p2, t p3) over F_q is ultimately periodic with least period π, the least period of the coefficient sequence of 1/p2; its periodic block is C_k = A^k C_0, with A a π×π circulant matrix generated by the coefficients of p3. Theorem 8 extends the picture to the 3-D array (p1/p2, t p3, p4): the orbit of periodic blocks in the k-th layer is M_4^k applied to the orbit in layer 0, M_4 a circulant built from p4. Theorem 10 adds that the sequence of preperiodic column partial sums is eventually periodic, and Corollary 11 exhibits a fami
Load-bearing premise
The whole description rests on the degree inequality d2 ≤ d1+1 and the coprime hypotheses: without them the clean identity C_k = A^k C_0 can fail (Example 5), and the proof that every column has period π depends on an implicit quotient-remainder reduction as numerator degree grows.
Editorial extensions
If this is right
- Each column k≥0 of the 2-D array (p1/p2, t p3) is ultimately periodic with the same least period π, independent of the column index, whenever d2 ≤ d1+1 and p2 is coprime to p1 and p3.
- The periodic block in column k is C_k = A^k C_0, so the entire collection of column blocks is the orbit of C_0 under A; since A has finite order, only finitely many distinct periodic blocks occur.
- In the 3-D array, the periodic-block orbit of layer k is M_4^k times the orbit of layer 0, so layer-to-layer evolution is governed entirely by the coefficients of p4.
- The preperiodic column partial sums S[k] form an eventually periodic sequence, and for the family with p1=1, p2=1+t, and even p3 whose coefficient sum is nonzero mod p, all S[k] vanish.
- Consequently, for any concrete array of this type, the full periodic structure of all columns can be written down from C_0 and A alone, without computing column by column.
Reading between the lines
- If the degree condition d2 ≤ d1+1 is dropped, Example 5 suggests the block law becomes C_k = P^{φ(k)} A^k C_0 with P a cyclic shift and φ(k) piecewise linear; one could test whether such a modified law holds generally outside the theorem's range.
- The claim that every high-index column still has period π leans on an implicit quotient-remainder step in Lemma 1 for numerators of degree exceeding d2; making that step explicit, or finding a counterexample, would settle whether the period-π assertion survives for all k.
- Because A and M_4 are circulant, the block-evolution law is essentially a linear recurrence on blocks; one might extend the same circulant-block analysis to other triangular arrays whose generating functions are rational, not just polynomial Riordan pairs.
- The vanishing partial-sums family suggests a mod-p combinatorial identity: for even p3, the first linear-length window of each column sums to zero; a combinatorial proof could illuminate why the preperiodic sums cancel.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies eventual periodicity in the columns of 2-D Riordan arrays (p1/p2, t p3) over a finite field F_q, and in the layers of 3-D arrays (p1/p2, t p3, p4). The main results are: Theorem 4, stating that, under d2 ≤ d1+1 and appropriate coprimality, every column k ≥ 1 is ultimately periodic with least period π (the least period of 1/p2), and that the periodic block evolves as C_k = A^k C_0 for a circulant matrix A built from the coefficients of p3; Theorem 8, giving an analogous orbit relation M_4^k between layers of a 3-D array; and Theorem 10, asserting that the preperiodic column partial sums form an eventually periodic sequence, with a special family (Corollary 11) for which they vanish. Examples illustrate the degree-condition restriction and show that A- and Z-sequences need not be periodic.
Significance. If established, the paper would give a clean, explicit description of column periodicity in terms of circulant matrices, extending earlier work on Riordan arrays over finite fields. The concrete finite-field examples and the explicit matrix formulas are valuable and make the intended statements easy to test. The main ideas are promising. However, two load-bearing proof gaps—one in Lemma 1 and one in the generating-function argument for Theorem 10—mean that the central claims are not fully justified as written. The gaps appear fixable, but they are not merely cosmetic.
major comments (3)
- [Section 2.2, Lemma 1] The proof of Lemma 1 silently treats only the case deg(p1) < deg(p2). The step 'b0 g(t) = p1(t)' after invoking Theorem 8.40 requires deg(g) < deg(f*) = deg(p2), which forces deg(p1) < deg(p2). The quotient-remainder reduction r(t)/p2(t) is mentioned in §2.2, but it is not used in the proof. This is load-bearing: under Theorem 4's hypothesis d2 ≤ d1+1, every column k≥1 has numerator p1(t)(t p3(t))^k of degree d1+k(d3+1) ≥ d2, so Lemma 1 is needed exactly in the case its proof omits. Please add an explicit argument that p1/p2 = q + r/p2 has the same least period as r/p2 and hence as 1/p2. The gap is fixable, and Example 5 does not contradict the statement, but the proof as written is incomplete.
- [Section 5, Theorem 10 proof, Eq. (5.6)] The equality S(x) = [t^D] R1(t) Σ (p3/t^{d3})^k x^k = [t^D] t^{d3} R1(t)/(t^{d3}-p3(t)x) requires a precise choice of expansion. The subsequent argument expands 1/Q in the t-adic f.p.s. topology because Q(0,x) ≠ 0, but this expansion does not in general compute the original S(x). For example, over F_2 take p1=1+t, p2=1+t+t^2, p3=1+t, so d1=1, d2=2, d3=1, D=-1; then R1=(1+t)/(1+t^3) and S[k] = [t^{-1}] R1(t)((1+t)/t)^k gives S(x)=x+x^2+x^4+x^5+... = (x+x^2)/(1-x^3). The t-adic coefficient [t^{-1}] of H(t,x)=t R1(t)/(t-(1+t)x), however, is 0. Thus the proof as written proves rationality of a different quantity. The theorem may be true, but it needs a justified diagonal/residue or constant-term argument that works with the x-adic/Laurent-in-t topology rather than the t-adic expansion of 1/Q.
- [Section 4, Theorem 8 proof, Eq. (4.11)] The displayed index in (4.11) is missing a factor d4. As written, the left side is [t^{(kd4+d1+1)-d2+m}] p1 p4^k/p2, but substituting p4^k = p4^{k-1} p4 gives a sum over z_j [t^{(kd4+d1+1)-d2+m-j}], and since kd4 = (k-1)d4 + d4, the index should be ((k-1)d4+d1+1)-d2 + d4 + m - j, not ((k-1)d4+d1+1)-d2 + m - j. The subsequent dot-product argument uses the corrected version, but the displayed equality is false as written and should be fixed.
minor comments (3)
- [Section 3.2, Theorem 4, Case 2] The last component of the vector in (3.4) is written as a sum over c_{d3+1-π-iπ}; the notation should be made unambiguous, e.g. c_{d3-(π-1)-iπ}, with the floor taken over the whole numerator. As typeset, the expression is hard to read.
- [References] Reference [10] spells the first author's name as 'Lidle'; it should be 'Lidl'.
- [Section 2.2, Lemma 2] In the proof, the notation 'n+1 = deg(q(t)) + 1' is clear, but the sentence before it refers to the (n+1)-st term without specifying zero-based indexing. A short clarification would help.
Circularity Check
No significant circularity: the column-block evolution C_k = A^k C_0 and the layer-orbit relation O(L_k)=M_4^k O(L_0) are derived from coefficient convolution, not assumed or fitted.
full rationale
The paper's central results are self-contained algebraic derivations. In Theorem 4, C_0 is the periodic block of the zeroth column (determined by p_1/p_2) and A is the circulant matrix built from the coefficients of p_3 and the period π of 1/p_2; the equality C_k = A^k C_0 is then proven by writing the k-th column generating function as (p_1 (t p_3)^k)/p_2 and applying coefficient extraction and periodicity of the base sequence. Nothing is fitted to the target block, and no quantity is defined in terms of the conclusion. Definition 7 introduces the orbit as {circ(v_3)^k C_0} only after Theorem 4 established that these are exactly the column periodic blocks, so Theorem 8's O(L_k)=M_4^k O(L_0) is an additional derived statement: the proof reduces it to showing C_{0,k}=M_4^k C_{0,0} by induction using the same convolution argument with p_4, not by construction. The citations to the authors' earlier papers [8,9] are motivational/historical (periodic partial sums in a special case), and the proofs of Theorem 4 and Theorem 8 rely on standard external facts [10,11], not on those self-citations. The paper explicitly flags the scope restriction d_2 ≤ d_1+1 in Example 5 and Note 2, and Corollary 11's zero-partial-sum result is a direct coefficient parity computation. There is a genuine proof-completeness concern in Lemma 1 when deg(p_1) ≥ deg(p_2) (the displayed argument uses deg(g)<deg(f*) to force b_0 g = p_1), which is exactly the regime needed for columns k≥1; however, this is a correctness gap, not a circularity, because the lemma's conclusion is not used as an input to itself and the later block-evolution formula remains a derived consequence rather than a renamed assumption. Under the standards for circularity, the derivation chain does not reduce to its own inputs.
Assumptions & free parameters
assumptions (4)
- standard math A rational generating function over F_q with non-zero denominator constant term is ultimately periodic (Lidl-Niederreiter, Theorem 8.40).
- standard math The least period of a linear recurring sequence over a finite field equals the order of its minimal polynomial (Lidl-Niederreiter, Theorem 8.44).
- domain assumption The Riordan group product, inverse, and coefficient extraction identities hold over F_q[[t]].
- domain assumption If gcd(p1,p2)=1, then after division p1 = q p2 + r with deg r < deg p2, the eventual periodic tail of p1/p2 is controlled by r/p2.
Cite this review
Pith. "Pith review of Periodicities in the Riordan arrays of polynomials over finite fields." pith.science (2026). https://pith.science/paper/PG6IFXGH
@misc{pith2026260713442,
author = {Pith},
title = {Pith review of: Periodicities in the Riordan arrays of polynomials over finite fields},
year = {2026},
howpublished = {\url{https://pith.science/paper/PG6IFXGH}},
note = {Machine review of arXiv:2607.13442}
}
abstract
We study periodicity properties of the 2-D $\bigl(p_1(t)/p_2(t),\, tp_3(t)\bigr)$ and 3-D $\bigl(p_1(t)/p_2(t),\, tp_3(t),\, p_4(t)\bigr)$ Riordan arrays over a finite field ${\mathbb F}_q$, where each $p_i(t)$ is a polynomial with $p_i(0)\neq 0$. We show that the columns of the 2-D Riordan array are eventually periodic sequences, where a circulant matrix generated by the coefficients of $p_3(t)$ determines the behavior of this periodicity as the column index grows indefinitely. Furthermore, we prove that the preperiodic column partial sums of the 2-D array are periodic, and present a family of the Riordan arrays for which such sequences of partial sums are identically zero. We also show that the layers of the 3-D Riordan array contain periodic orbits related to each other via powers of a circulant matrix generated by the coefficients of $p_4(t)$.
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Reference graph
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Reviewed August 2, 2026 · model on record in the stance chip above.
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