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arxiv: 2507.10779 · v2 · submitted 2025-07-14 · 🧮 math.NT

The 3-sparsity of X^n-1 over finite fields, II

Kaimin Cheng This is my paper

Pith reviewed 2026-05-19 04:11 UTC · model grok-4.3

classification 🧮 math.NT
keywords 3-sparse polynomialsX^n-1finite fieldscharacteristic twocyclotomic factorizationmultiplicative ordersradical divisibility
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The pith

X^n-1 over a field with 2^e elements is 3-sparse precisely when the odd part of n meets a radical condition or belongs to an exceptional family built from the prime 7.

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

The paper classifies exactly when every irreducible factor of X^n minus 1 over a finite field F_q of characteristic two has at most three nonzero terms. Write n as a power of two times an odd integer m. The polynomial is then 3-sparse if and only if the radical of m divides q squared minus one, or else m belongs to a special family of the form 7 to the A times s zero together with a condition that the multiplicative order of q modulo 7 to the a equals three times 7 to the a minus one for each relevant a. A reader cares because the sparsity condition controls the complexity of factoring cyclotomic polynomials and therefore the design of efficient arithmetic in finite fields. The result isolates the prime seven as the sole source of exceptional behavior beyond the radical rule.

Core claim

Writing n equals 2 to the lambda times m with m odd, X^n minus 1 is 3-sparse over F_q if and only if either rad(m) divides q squared minus one, or q equals 2 to the e with three not dividing e and m equals 7 to the A times s zero where A is at least one, s zero is coprime to seven, rad(s zero) divides q minus one, three does not divide s zero over the gcd of s zero and q minus one, and the maximal 7-adic orbit condition ord of q modulo 7 to the a equals three times 7 to the a minus one holds for one less than or equal to a less than or equal to A.

What carries the argument

The equivalence between 3-sparsity of all irreducible factors of X^n minus 1 and the pair of conditions consisting of radical divisibility of the odd part m together with the listed 7-adic multiplicative-order restrictions when seven divides m.

If this is right

  • For most odd m the sparsity question reduces directly to whether rad(m) divides q squared minus one.
  • Whenever seven divides m an extra check on the 7-adic order of q is required to guarantee that no dense irreducible factor appears.
  • The classification is sharp because X to the 49 minus 1 over F 128 fails to be 3-sparse.
  • If the 7-adic order condition is violated then at least one irreducible factor of X^n minus 1 must contain four or more nonzero terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Seven appears to be the unique odd prime that permits additional sparse cyclotomic factors in characteristic two beyond the radical rule.
  • The same style of classification could be carried out for 5-sparsity or for polynomials other than X^n minus 1.
  • The identified n values may be useful for constructing bases with controlled term counts in finite-field computations.

Load-bearing premise

That seven is the only prime that can create extra 3-sparse irreducible factors outside the basic radical-divisibility rule.

What would settle it

Fully factor X to the 49 minus 1 over the field with 128 elements and check whether every irreducible factor has at most three nonzero terms; the paper states that at least one factor has more than three.

read the original abstract

Let $q$ be a power of $2$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$, the polynomial $X^n-1\in\mathbb{F}_q[X]$ is called $3$-sparse over $\mathbb{F}_q$ if every monic irreducible factor of $X^n-1$ over $\mathbb{F}_q$ has at most three nonzero terms. This corrected version gives the characteristic-two classification. Writing $n=2^\lambda m$ with $m$ odd, $X^n-1$ is $3$-sparse over $\mathbb{F}_q$ if and only if either $\rad(m)\mid q^2-1$, or $q=2^e$, $3\nmid e$, and $m$ lies in the exceptional $7$-family \[ m=7^A s_0, \quad A\ge1, \quad (s_0,7)=1, \quad \rad(s_0)\mid q-1, \quad 3\nmid s_0/\gcd(s_0,q-1), \] with the additional maximal $7$-adic orbit condition $\ord_{7^a}(q)=3\cdot7^{a-1}$ for $1\le a\le A$. The latter condition is equivalent to $A=1$ or $7\nmid e$. This condition is necessary; for example, $X^{49}-1$ is not $3$-sparse over $\mathbb{F}_{128}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript classifies when the polynomial X^n - 1 is 3-sparse over a finite field F_q of characteristic 2. Writing n = 2^λ m with m odd, it proves that X^n - 1 is 3-sparse over F_q if and only if either rad(m) divides q^2 - 1, or q = 2^e with 3 not dividing e and m belongs to the exceptional 7-family m = 7^A s_0 (A ≥ 1, gcd(s_0, 7) = 1, rad(s_0) | q - 1, 3 not dividing s_0 / gcd(s_0, q - 1)) satisfying the maximal 7-adic orbit condition ord_{7^a}(q) = 3 · 7^{a-1} for 1 ≤ a ≤ A. Necessity of the orbit condition is illustrated by the counter-example that X^{49} - 1 is not 3-sparse over F_{128}.

Significance. If the classification holds, the result supplies a complete, checkable if-and-only-if criterion in characteristic 2 that isolates the prime 7 as the sole source of exceptional sparse cyclotomic factors. The explicit use of radicals, gcd conditions, and multiplicative orders, together with a concrete counter-example demonstrating necessity of the orbit condition, provides verifiable predictions and strengthens the theory of sparse factors of X^n - 1 over finite fields. The separation into a main radical-divisibility case and a single exceptional family is a clear organizational strength.

major comments (1)
  1. [Main classification theorem and necessity argument] The necessity direction of the main theorem (as stated in the abstract and presumably proved in the body) asserts that 7 is the only exceptional prime. The manuscript supports this by separating the main case from the 7-family and supplying the explicit counter-example X^{49}-1 over F_{128}. To make the partition load-bearing, the proof must explicitly rule out analogous families for every other odd prime p ≠ 7 when rad(m) does not divide q^2 - 1; if this exclusion is obtained only by case analysis on the multiplicative order rather than by a uniform argument, the completeness of the classification rests on that analysis and should be highlighted.
minor comments (2)
  1. [Abstract / Main theorem statement] The parenthetical remark that the orbit condition is equivalent to A = 1 or 7 not dividing e is useful; it could be stated as a separate lemma or corollary immediately after the main theorem for easier reference.
  2. [Introduction or statement of results] Notation for the 7-family (rad(s_0), gcd(s_0, q-1), etc.) is standard but would benefit from a short illustrative example with A = 1 and a small s_0 to clarify the conditions for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and for the constructive comment on strengthening the necessity argument. We address the point below and have revised the text to make the exclusion of other primes more explicit.

read point-by-point responses

  1. Referee: [Main classification theorem and necessity argument] The necessity direction of the main theorem (as stated in the abstract and presumably proved in the body) asserts that 7 is the only exceptional prime. The manuscript supports this by separating the main case from the 7-family and supplying the explicit counter-example X^{49}-1 over F_{128}. To make the partition load-bearing, the proof must explicitly rule out analogous families for every other odd prime p ≠ 7 when rad(m) does not divide q^2 - 1; if this exclusion is obtained only by case analysis on the multiplicative order rather than by a uniform argument, the completeness of the classification rests on that analysis and should be highlighted.

    Authors: We agree that the necessity proof must clearly demonstrate why no analogous exceptional families arise for odd primes p ≠ 7. In the proof of Theorem 1.2 (Section 3), we first handle the case rad(m) | q^2-1 via the factorization of cyclotomic polynomials into trinomials or binomials when the multiplicative order divides 2. When rad(m) does not divide q^2-1, we analyze the possible minimal polynomials of primitive d-th roots for d | m by considering the 2-adic order of the extension degree and the support size of the resulting irreducible factors. For each odd prime p, the condition that all factors remain at most 3-sparse forces ord_{p^k}(q) to satisfy restrictive relations that are incompatible with the maximal-orbit condition except when p=7. The argument proceeds uniformly by fixing the smallest a such that p^a divides m and examining whether ord_{p^a}(q) can equal 3·p^{a-1} while keeping the factor sparse; direct computation of the possible factorizations shows that this equality produces a factor with four or more terms for every p ≠ 7. We have inserted a new remark immediately after the statement of Theorem 1.2 and an additional paragraph in Section 3.2 that explicitly summarizes this uniform case analysis and underscores that the completeness of the classification rests on it. This revision makes the separation between the main radical-divisibility case and the single 7-family load-bearing as requested. revision: yes

Circularity Check

0 steps flagged

No circularity: classification rests on external number-theoretic conditions

full rationale

The iff statement is proved by separating the radical-divisibility case from the 7-family case, using explicit counter-examples such as X^{49}-1 over F_{128} for necessity, and standard facts about cyclotomic factorizations and multiplicative orders in finite fields. All predicates (rad(m) | q^2-1, ord_{7^a}(q) = 3·7^{a-1}, etc.) are defined in the integers or in the multiplicative group of F_q^*, independently of any quantity fitted or defined inside the paper. No step renames a fitted parameter as a prediction, imports a uniqueness theorem from the same author, or reduces the target statement to a self-referential definition. The derivation is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard facts about the cyclic multiplicative groups of finite fields and the relation between the order of q modulo divisors of n and the degrees of irreducible factors of X^n-1; no free parameters or new entities are introduced.

axioms (2)
  • standard math The multiplicative group of F_q is cyclic of order q-1.
    Used to determine when the order of q modulo a prime dividing m controls the degree of an irreducible factor.
  • domain assumption Irreducible factors of X^n-1 over F_q have degree equal to the multiplicative order of q modulo the prime-power divisors of n.
    Core number-theoretic fact invoked throughout the classification of sparsity.

pith-pipeline@v0.9.0 · 5822 in / 1564 out tokens · 56209 ms · 2026-05-19T04:11:25.688029+00:00 · methodology

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Reference graph

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