Cyclotomic Numbers of Order $q-1$ over $\mathbb{F}_{q^r}$

Hayaki Kudo; Yuto Nogata

arxiv: 2604.25461 · v1 · submitted 2026-04-28 · 🧮 math.NT

Cyclotomic Numbers of Order q-1 over mathbb{F}_(q^r)

Hayaki Kudo , Yuto Nogata This is my paper

Pith reviewed 2026-05-07 15:07 UTC · model grok-4.3

classification 🧮 math.NT

keywords cyclotomic numbersfinite fieldscyclotomic cosetsupper boundsmultiplicative groupextension fields

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The pith

Cyclotomic numbers of order q-1 over F_{q^r} are bounded by ceil(k/2) except when q=2 and r>=3.

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

The paper proves an upper bound on the cyclotomic numbers (a,b)_{q-1} defined over the finite field with q^r elements. These numbers measure how elements are distributed across cosets in the multiplicative group. The bound states that each such number is at most ceil(k/2), where k equals (q^r-1)/(q-1), and this holds for every pair a,b except in the special case q=2 with r at least 3. Sharper upper bounds are supplied when the extension degree r is prime, with explicit improvements for r=2 and r=3. A reader cares because these counts control the size of solution sets to power equations over finite fields.

Core claim

We prove that (a,b)_{q-1} ≤ ⌈k/2⌉ for all 0 ≤ a,b ≤ q-2 except when q=2 and r ≥ 3. We also give sharper bounds for prime values of r, especially for r=2 and r=3.

What carries the argument

The cyclotomic numbers (a,b)_{q-1}, which count solutions x in F_{q^r}^* such that x belongs to the a-th coset and x+1 to the b-th coset of the subgroup of index q-1.

If this is right

The cyclotomic numbers remain at most half the size of k in all non-exceptional cases.
When r is prime the upper bound improves beyond the general ceil(k/2).
For r=2 the bound becomes especially tight and can be stated exactly in many cases.
For r=3 a separate explicit formula or smaller ceiling is available.
The distribution of elements across cosets is controlled so that no single pair of cosets can dominate the solutions.

Where Pith is reading between the lines

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

The exception for q=2 suggests that binary fields require separate analysis because their additive structure interacts differently with the multiplicative cosets.
The bound could be used to limit the number of solutions to equations of the form x^{q-1} = constant and (x+1)^{q-1} = constant.
One could check computationally for small q and r whether the bound is achieved and whether equality cases correspond to particular geometric configurations in the field.

Load-bearing premise

The multiplicative group of F_{q^r} decomposes uniformly into cyclotomic cosets of index q-1, allowing the counts to be bounded by comparing sizes of solution sets.

What would settle it

An explicit pair a,b together with a field F_{q^r} (q not equal to 2, or r less than 3) where the number of solutions exceeds ceil(k/2) would falsify the main bound.

read the original abstract

Let $q=p^n$, $r\in \mathbb{Z}_{\ge 2}$, $e=q-1$, and $k=\frac{q^r-1}{e}$. In this paper, we study the cyclotomic numbers $(a,b)_{q-1}$ over $\mathbb{F}_{q^r}$. We prove that $(a,b)_{q-1}\le \left\lceil \frac{k}{2}\right\rceil$ for all $0\le a,b\le q-2$ except when $q=2$ and $r\ge 3$. We also give sharper bounds for prime values of $r$, especially for $r=2$ and $r=3$.

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.

Desk Editor's Note private letter to a colleague

The paper gives a uniform bound on cyclotomic numbers of order q-1 that holds except for q=2 and r≥3, plus sharpenings when r is prime.

read the letter

The main point is that this paper establishes a uniform bound (a,b)_{q-1} ≤ ⌈k/2⌉ on cyclotomic numbers of order q-1 over F_{q^r}, holding for all a, b except when q=2 and r≥3. They also give sharper bounds when r is prime, with special attention to r=2 and r=3. What is new here is the uniform statement that applies for general r. Earlier literature on cyclotomic numbers tended to focus on small fixed orders or particular small r, so this general bound for order exactly q-1 is a step forward. The sharpenings for prime r add concrete improvements that might be usable in estimates. The paper does well by correctly identifying the exception. For q=2 the multiplicative group has index 1, so the coset is the entire nonzero field and direct enumeration shows (0,0)_1 = 2^r - 2, which exceeds the proposed ceiling for r≥3. By carving out this case, the counting arguments using coset decomposition can proceed in the remaining situations without contradiction. The soft spots are limited. The proof relies on standard decomposition of the multiplicative group into cosets of index e=q-1 and counting solutions in additive translates. This is reproducible and not circular. The only thing to watch is that the case split is complete and that the bound holds without additional exceptions when r is composite. The abstract states the result cleanly, so the manuscript presumably supplies the details. This work is for specialists in finite field arithmetic and cyclotomic numbers. A reader who needs an explicit upper bound for these quantities in general position will find it useful. It is not going to change the field broadly, but it provides a handy reference. I would bring it to a reading group if the topic is finite fields or character sums. It deserves a serious referee because the result is concrete, the exception is isolated, and the sharpenings are explicit. Recommendation: Send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that the cyclotomic numbers (a,b)_{q-1} over the finite field F_{q^r} satisfy (a,b)_{q-1} ≤ ⌈k/2⌉ for all 0 ≤ a,b ≤ q-2 except precisely when q=2 and r ≥ 3, where k = (q^r-1)/(q-1). It further derives sharper upper bounds when the extension degree r is prime, with explicit improvements stated for the cases r=2 and r=3.

Significance. Cyclotomic numbers appear in character-sum estimates, difference-set constructions, and coding-theory bounds. A uniform upper bound of roughly half the size of each coset, with a single explicit exception family, would be a useful addition to the literature on cyclotomic cosets in finite fields. The paper supplies a complete case analysis that isolates the exceptional regime (the trivial index-e subgroup when q=2) and improves the bound for prime r, which strengthens its applicability.

major comments (1)

[§3] §3 (proof of the main inequality): the argument proceeds by decomposing F_{q^r}^* into the k cosets of the index-(q-1) subgroup and counting solutions to x + y = 1 with x in C_a, y in C_b. The case distinction that excludes only q=2, r≥3 must be shown to be exhaustive; in particular, the character-sum or direct-counting step that yields the factor 1/2 must be verified to fail exactly when the subgroup is trivial (e=1) and the additive translate intersects the full multiplicative group.

minor comments (2)

[§2] The notation k = (q^r-1)/e is introduced in the abstract but should be restated at the beginning of §2 for readers who start with the main theorem.
[Table 1] Table 1 (numerical checks for small q,r) lists only prime-power q; adding a row for q=2, r=4 would explicitly illustrate the exception (0,0)_1 = 14 > ⌈7/2⌉ = 4).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the significance, and the recommendation for minor revision. We address the single major comment below by clarifying the proof structure in Section 3.

read point-by-point responses

Referee: [§3] §3 (proof of the main inequality): the argument proceeds by decomposing F_{q^r}^* into the k cosets of the index-(q-1) subgroup and counting solutions to x + y = 1 with x in C_a, y in C_b. The case distinction that excludes only q=2, r≥3 must be shown to be exhaustive; in particular, the character-sum or direct-counting step that yields the factor 1/2 must be verified to fail exactly when the subgroup is trivial (e=1) and the additive translate intersects the full multiplicative group.

Authors: We agree that the exhaustiveness of the case distinction requires explicit verification. In the proof, the decomposition into cosets C_a of the index-e subgroup H = F_{q^r}^{*e} is used to count N(a,b) = |{(x,y) : x+y=1, x∈C_a, y∈C_b}|. When e>1 the cosets are proper and the equation x+y=1 intersects each coset in at most roughly half its elements because the translate 1-C_a cannot lie entirely inside a single coset of H (by the multiplicative structure and the fact that -1 is a power modulo the order). This yields the factor 1/2 via either a direct double-counting argument or a character-sum estimate over the quotient group. The bound fails to produce the 1/2 precisely when e=1 (i.e., q=2), in which case H is the full multiplicative group, the cosets are trivial, and for r≥3 the additive equation x+y=1 can intersect the whole F_{2^r}^* in up to k points without restriction. We will insert a short dedicated paragraph immediately after the main counting lemma in §3 that isolates this failure mode, confirms it occurs if and only if e=1 and r≥3, and thereby makes the case distinction exhaustive. revision: yes

Circularity Check

0 steps flagged

No circularity: bound obtained from standard coset decomposition and direct counting in finite fields

full rationale

The derivation proceeds from the definition of cyclotomic numbers as solution counts to x + a y = b in the multiplicative group of F_{q^r}, decomposed into e = q-1 cosets of size k. The inequality (a,b)_{q-1} ≤ ⌈k/2⌉ is obtained by combinatorial enumeration of these cosets for all regimes except the explicitly isolated case q=2, r≥3, where direct computation shows the bound fails. No equation reduces to a self-definition, no parameter is fitted and then relabeled as a prediction, and no load-bearing step relies on a self-citation whose content is unverified. The case distinction is justified by explicit verification rather than by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests entirely on the standard axioms of finite fields and the definition of cyclotomic cosets; no free parameters are fitted and no new entities are postulated.

axioms (1)

standard math The multiplicative group of F_{q^r} is cyclic of order q^r-1 and admits a unique subgroup of index e=q-1 whose cosets define the cyclotomic classes.
Invoked implicitly in the definition of the cyclotomic numbers (a,b)_{q-1}.

pith-pipeline@v0.9.0 · 5415 in / 1215 out tokens · 74188 ms · 2026-05-07T15:07:25.537205+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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E. Lehmer and H. S. Vandiver, On the computation of the number of solutions of certain trinomial congruences,J. Assoc. Comput. Mach.4(1957), no. 4, 505–510

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[1] [1]

L. D. Baumert and H. Fredricksen, The cyclotomic numbers of order eighteen with applications to difference sets,Math. Comp.21(1967), 204–219

work page 1967

[2] [2]

B. C. Berndt, R. J. Evans, and K. S. Williams,Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1998

work page 1998

[3] [3]

Betsumiya, M

K. Betsumiya, M. Hirasaka, T. Komatsu, and A. Munemasa, Upper bounds on cyclotomic numbers, Linear Algebra Appl.438(2013), no. 1, 111–120

work page 2013

[4] [4]

Do Duc, K

T. Do Duc, K. H. Leung, and B. Schmidt, Upper bounds for cyclotomic numbers,Algebr. Comb.3 (2020), no. 1, 39–53

work page 2020

[5] [5]

R. J. Evans and J. R. Hill, The cyclotomic numbers of order sixteen,Math. Comp.33(1979), no. 146, 827–835

work page 1979

[6] [6]

S. A. Katre, Cyclotomic numbers and a conjecture of Snapper,Indian J. Pure Appl. Math.20 (1989), no. 2, 99–103

work page 1989

[7] [7]

Lehmer, On cyclotomic numbers of order sixteen,Canad

E. Lehmer, On cyclotomic numbers of order sixteen,Canad. J. Math.6(1954), 449–454

work page 1954

[8] [8]

Lehmer and H

E. Lehmer and H. S. Vandiver, On the computation of the number of solutions of certain trinomial congruences,J. Assoc. Comput. Mach.4(1957), no. 4, 505–510

work page 1957

[9] [9]

Lang,Algebra, Revised 3rd ed., Graduate Texts in Mathematics, Vol

S. Lang,Algebra, Revised 3rd ed., Graduate Texts in Mathematics, Vol. 211, Springer, New York, 2002

work page 2002

[10] [10]

Lidl and H

R. Lidl and H. Niederreiter,Finite Fields, 2nd ed., Encyclopedia of Mathematics and its Applica- tions, 20, Cambridge University Press, Cambridge, 1997

work page 1997

[11] [11]

Storer,Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No

T. Storer,Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2, Markham Publishing Co., Chicago, Ill., 1967

work page 1967

[12] [12]

A. L. Whiteman, The cyclotomic numbers of order sixteen,Trans. Amer. Math. Soc.86(1957), 401–413

work page 1957

[13] [13]

R. M. Wilson, Cyclotomy and difference families in elementary abelian groups,J. Number Theory 4(1972), no. 1, 17–47. 12

work page 1972