pith. sign in

arxiv: 1907.09355 · v1 · pith:UM7TGDTXnew · submitted 2019-07-22 · 🧮 math.NT

Permutation Binomials over Finite Fields

Jos\'e Alves Oliveira , F. E. Brochero Mart\'inez This is my paper

Pith reviewed 2026-05-24 17:51 UTC · model grok-4.3

classification 🧮 math.NT
keywords permutation polynomialsfinite fieldsbinomial polynomialsalgebraic curvesrational pointsnumber of solutions
0
0 comments X

The pith

The exact number of a in F_q for which x^n(x^{(q-1)/r} + a) is a permutation polynomial equals the number of rational points on associated curves, for r=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 determines the precise count of elements a in a finite field that make the given binomial a permutation polynomial when the second exponent is (q-1)/2 or (q-1)/3. Permutation polynomials are the functions that hit every field element exactly once, so counting them matters for applications that rely on bijective maps over finite fields. The authors reduce the question of when the binomial permutes the field to the problem of counting solutions to certain polynomial equations. Those equations define algebraic curves, and the permutation count is then read off from the number of rational points on the curves.

Core claim

We use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements a in F_q for which the binomial x^n(x^{(q-1)/r} + a) is a permutation polynomial in the cases r=2 and r=3.

What carries the argument

The reduction of the permutation condition for the binomial to the number of rational points on associated algebraic curves over F_q.

If this is right

  • The count of good a is obtained without testing every candidate a individually.
  • Explicit formulas or values for the number of such a follow once the curve point counts are known for the two cases.
  • The same reduction technique applies uniformly to both the r=2 and r=3 families of binomials.

Where Pith is reading between the lines

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

  • The curve-point method may yield closed-form expressions in q once the point counts are evaluated using existing formulas for the specific curves that arise.
  • Similar reductions could be attempted for other fixed r or for trinomials, though the resulting curves would generally be of higher genus.

Load-bearing premise

The permutation property of the given binomial is equivalent to (and can be exactly counted via) the number of rational points on associated algebraic curves over F_q.

What would settle it

For a concrete small field such as q=13 and r=2, direct enumeration of all a and direct computation of the number of points on the corresponding curve should disagree if the claimed equivalence fails.

read the original abstract

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for which the binomial $x^n(x^{(q-1)/r} + a)$ is a permutation polynomial in the cases $r = 2$ 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.

Referee Report

0 major / 3 minor

Summary. The manuscript determines the exact number of elements a in F_q such that the binomial x^n (x^{(q-1)/r} + a) is a permutation polynomial over F_q, for the cases r=2 and r=3. The method associates the permutation criterion with the number of F_q-rational points on associated algebraic curves.

Significance. If the derivations are correct, the paper supplies exact (non-asymptotic) counts for these specific r values using a standard technique from the theory of permutation polynomials over finite fields. This strengthens the literature on value sets and permutation binomials, which are relevant to coding theory and cryptography; the approach via curve point counts is rigorous when the reduction is valid.

minor comments (3)
  1. The abstract and introduction should include a brief statement of the main theorems (e.g., the explicit formulas for the counts when r=2 and r=3) to make the central results immediately visible.
  2. Notation for the exponent n and the parameter r should be clarified early; it is not immediately clear whether n is fixed or varies with q.
  3. Any explicit formulas or tables giving the counts as functions of q should be cross-referenced to the corresponding curve-point formulas in the body of the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the exact count of suitable a by relating the permutation polynomial condition for the given binomial (when r=2 or 3) to the number of F_q-rational points on associated algebraic curves. This is a standard, externally grounded technique in the literature on finite fields (via character sums or value-set criteria rewritten as curve equations) and does not reduce to any self-definitional equivalence, fitted parameter renamed as prediction, or load-bearing self-citation. The central claim remains independent of the paper's own inputs and is self-contained against external algebraic-geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the unelaborated equivalence between the permutation condition and rational-point counts on curves; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Permutation property of the binomial is equivalent to the number of rational points on associated algebraic curves
    Explicitly invoked in the abstract as the method used to obtain the counts.

pith-pipeline@v0.9.0 · 5589 in / 1060 out tokens · 25275 ms · 2026-05-24T17:51:13.667654+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Akbary and W

    A. Akbary and W. Qiang, A Generalized Lucas Sequence and Permutation Binomials , Proc. Amer. Math. Soc. 134 (2006), pp. 15-22

    work page 2006

  2. [2]

    Bhattacharya, S

    S. Bhattacharya, S. Sarkar, On Some Permutation Binomials and Trinomials over F2n, Designs, Codes and Cryptography, vol. 82, Issue 1-2 (2017), pp. 149-16 0

    work page 2017

  3. [3]

    Bhattacharya, S

    S. Bhattacharya, S. Sarkar, and A. C ¸ e¸ smelio˘ glu On Some Permutation Binomials of the Form x 2n −1 k +1+ax over F2n : Existence and Count , Arithmetic of Finite Fields, Lecture Notes in Computer Science, vol 7369 (2012), pp. 236-246

    work page 2012

  4. [4]

    Carlitz, Some Theorems on Permutation Polynomials , Bull

    L. Carlitz, Some Theorems on Permutation Polynomials , Bull. Amer. Math. Soc. vol 68 (2) (1962), pp. 120-122

    work page 1962

  5. [5]

    Chu and S

    W. Chu and S. W. Golomb, Circular Tuscan- k Arrays From Permutation Binomials , J. Comb. Theory A 97 (2002), 195-202

    work page 2002

  6. [6]

    C. J. Colbourn, T. Klove and A. C. H. Ling, Permutation Arrays for Powerline Communication and Mutually Orthogonal Latin Squares , IEEE Trans. Inf. Theory 50 (2004), pp. 1289-1291

    work page 2004

  7. [7]

    J. F. Dillon and H. Dobbertin, New Cyclic Difference Sets with Singer Parameters , Finite Fields Appl. 10 (2004), pp. 342-389

    work page 2004

  8. [8]

    L. E. Dickson, The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group , Annals of Math, 11 (1896), pp. 65-120

    work page

  9. [9]

    G. H. Hardy, E. M. Wright, R. Heath-Brown, J. Silverman, A. Wiles , An introduction to the Theory of Numbers (6th ed.). Oxford Science Publications, 2008

    work page 2008

  10. [10]

    Hermite, Sur Les Fonctions de Sept Lettres , C

    C. Hermite, Sur Les Fonctions de Sept Lettres , C. R. Acad. Sci. Paris 57 (1863), pp. 750-757

    work page

  11. [11]

    S. Y. Kim and J. B. Lee, Permutation Polynomials of the Type x q−1 m +1 +ax, Commun. Korean Math. Soc. 10 (1995), pp. 823-829

    work page 1995

  12. [12]

    K. Li, L. Qu, X.Chen, New Classes of Permutation Binomials and Permutation Trino mials over Finite Fields , Finite Fields and Their Applications, vol. 43 (2017), pp. 69-85

    work page 2017

  13. [13]

    Lidl and H

    R. Lidl and H. Niederreiter, Finite Fields , vol. 20, Cambridge University Press, 1997. PERMUTATION BINOMIALS OVER FINITE FIELDS 13

    work page 1997

  14. [14]

    Masuda and M

    A. Masuda and M. Zieve, Permutation Binomials over Finite Fields , Transactions of the American Mathematical Society, 361 (2009), pp. 4169-4180

    work page 2009

  15. [15]

    Niederreiter and K

    H. Niederreiter and K. H. Robinson, Complete Mappings of Finite Fields , J. Australian Math. Soc. Ser. A, vol. 33 (1982), pp. 197-212

    work page 1982

  16. [16]

    Niven, Irrational Numbers, Carus Math

    I. Niven, Irrational Numbers, Carus Math. Monographs 11, MAA, 1956

    work page 1956

  17. [17]

    Y. H. Park and J. B. Lee, Permutation Polynomials and Group Permutation Polynomial s, Bull. Austral. Math. Soc. 63 (2001), 67-74

    work page 2001

  18. [18]

    5.1), The Sage Developers, 2017, https://www.sagemath.org

    SageMath, the Sage Mathematics Software System (Version 7. 5.1), The Sage Developers, 2017, https://www.sagemath.org

    work page 2017

  19. [19]

    Small, Permutation Binomials , Internat

    C. Small, Permutation Binomials , Internat. J. Math. & Math. Sci. 13 (1990), pp. 337-342

    work page 1990

  20. [20]

    J. H. Silverman, The Arithmetic of Elliptic Curves , vol. 106, Springer Science & Business Media, 2009

    work page 2009

  21. [21]

    J. Sun, O. Y. Takeshita and M. P. Fitz, Permutation Polynomial Based Deterministic Interleavers for Turbo Codes, Proceedings 2003 IEEE International Symposium on Information Theory (Yokohama, Japan, 2003), 319

    work page 2003

  22. [22]

    Wan, Permutation Binomials over Finite Fields , Acta Math

    D. Wan, Permutation Binomials over Finite Fields , Acta Math. Sinica (New Series) 10 (1994), pp. 30-35

    work page 1994

  23. [23]

    Wan, The Number of Permutation Polynomials of the Form f (x) + cx over a Finite Field , Pro- ceedings of the Edinburgh Mathematical Society (Series 2), vol 38 (1993), pp

    D. Wan, The Number of Permutation Polynomials of the Form f (x) + cx over a Finite Field , Pro- ceedings of the Edinburgh Mathematical Society (Series 2), vol 38 (1993), pp. 133-149

    work page 1993

  24. [24]

    Wan and R

    D. Wan and R. Lidl, Permutation Polynomials of the Form xrf (x(q−1)/d) and Their Group Struc- ture, Monatshefte f¨ ur Mathematik, 112 (1991), pp. 149-163

    work page 1991

  25. [25]

    Wang, On Permutation Polynomials , Finite Fields Appl

    L. Wang, On Permutation Polynomials , Finite Fields Appl. 8 (2002), pp. 311-322

    work page 2002

  26. [26]

    Xing-D. Hou, S. D. Lappano, Determination of a Type of Permutation Binomials over Finit e Fields , Journal of Number Theory, vol. 147 (2015), pp. 14-23

    work page 2015

  27. [27]

    Zieve, On some permutation polynomials over Fq of the form xrh(x(q−1)/d)

    M. Zieve, On some permutation polynomials over Fq of the form xrh(x(q−1)/d). Proc. Amer. Math. Soc. 137 (2009), 2209-2216. Departamento de Matem ´atica, Universidade Federal de Minas Gerais, UFMG, Belo Horizonte, MG, 31270-901, Brazil, E-mail address : jose-alvesoliveira@hotmail.com E-mail address : fbrocher@mat.ufmg.br

    work page 2009