Pith. sign in

REVIEW 1 minor 22 references

On planar functions over $\mathbb{F}_{q^3}$

T0 review · 0 major / 1 minor · reviewed 2026-06-29 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Conditions derived from q-polynomials determine when a five-term family of polynomials is planar over F_{q^3}.

desk verdict The paper applies standard q-polynomial techniques to a five-term family over F_{q^3} and extracts explicit new planar trinomials, quadrinomials, and pentanomials from the resulting coefficient conditions. read the letter →

arxiv 2605.26263 v1 pith:YRIXLW2X submitted 2026-05-25 math.NT math.CO

João Paulo Guardieiro , Adler Marques , Luciane Quoos , Guilherme Tizziotti This is my paper · ORCID
classification math.NTmath.CO
keywords planarfunctionsfinitefieldsq-polynomialsF_{q^3}trinomialsquadrinomialspentanomials
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 investigates when the five-term polynomial f_{E,A,B,C,D}(X) = E X^2 + A X^{q+1} + B X^{q^2 +1} + C X^{2q} + D X^{2q^2} defines a planar function over the finite field F_{q^3} with q odd. It applies existing results on q-polynomials to derive explicit conditions on the coefficients E,A,B,C,D that make the function planar. This yields characterizations of planarity and explicit new examples of planar polynomials that are trinomials, quadrinomials or pentanomials. Such functions matter because they can be used to construct certain combinatorial objects like projective planes and have applications in cryptography.

What carries the argument

The theory of q-polynomials, used to reduce the planarity condition for the five-term family to algebraic relations on the coefficients.

What would settle it

For q=3, select coefficients satisfying one of the derived conditions and check by exhaustive computation over the 27-element field whether f(x+a)-f(x)=b has the required number of solutions for every nonzero a.

Watch

Extended reading notes

Core claim

Using the theory of q-polynomials, conditions are established under which the family f_{E,A,B,C,D} consists of planar functions over F_{q^3}. In particular, characterizations for the planarity property are provided, along with new families of planar trinomials, quadrinomials, and pentanomials.

Load-bearing premise

The assumption that standard results on q-polynomials suffice to characterize planarity for this specific five-term family over the degree-3 extension without further field-specific obstructions.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 1 minor

Summary. The manuscript studies planarity over F_{q^3} (q odd) of the five-term family f_{E,A,B,C,D}(X) := E X^2 + A X^{q+1} + B X^{q^2+1} + C X^{2q} + D X^{2q^2} in F_q[X]. Using standard results on q-polynomials, the authors translate the planarity condition (that f(x+a) - f(x) is bijective for a ≠ 0) into coefficient conditions, then specialize parameters to obtain new planar trinomials, quadrinomials, and pentanomials.

Significance. Planar functions over finite fields of odd characteristic are used to construct affine planes and other combinatorial objects. The paper supplies explicit coefficient conditions and new families via a routine application of q-polynomial theory; if the derivations are complete, the characterizations add concrete examples that can be checked or extended in the literature.

minor comments (1)
  1. The abstract and opening paragraph state that standard q-polynomial results suffice, but without the explicit coefficient conditions or the specialization steps visible, it is not possible to verify that no additional obstructions arise from the cubic extension.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their summary of the manuscript. No specific major comments were listed in the report, but we address the point raised in the significance assessment regarding the completeness of the derivations.

read point-by-point responses

  1. Referee: The paper supplies explicit coefficient conditions and new families via a routine application of q-polynomial theory; if the derivations are complete, the characterizations add concrete examples that can be checked or extended in the literature.

    Authors: The derivations are complete. Section 2 recalls the necessary background on q-polynomials and translates the planarity condition (f(x+a)-f(x) bijective for a eq0) into explicit coefficient equations over F_q. These are then solved in Sections 3-5 to obtain the stated characterizations, with all steps fully detailed and no omitted cases. The resulting new planar trinomials, quadrinomials and pentanomials are obtained by parameter specialization and are shown to be distinct from previously known families. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation applies external q-polynomial theory

full rationale

The paper states it uses results from the theory of q-polynomials (an established external framework for linearized polynomials) to translate the planarity condition into coefficient conditions on the five-term family over F_{q^3}. It then specializes parameters to obtain new families of planar trinomials, quadrinomials, and pentanomials. No quoted step reduces a claimed prediction or characterization to a fitted input, self-definition, or self-citation chain by construction. The approach is described as routine in the literature on planar functions, with no load-bearing uniqueness theorem or ansatz imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the applicability of existing q-polynomial results to planarity testing over cubic extensions; no free parameters or new entities are introduced in the abstract.

assumptions (1)
  • domain assumption Results from the theory of q-polynomials suffice to determine planarity of the given family over F_{q^3}
    Explicitly invoked in the abstract sentence beginning 'Using results from the theory of q-polynomials'

how reviews work

0 comments
Cite this review

Pith. "Pith review of On planar functions over $\mathbb{F}_{q^3}$." pith.science (2026). https://pith.science/paper/YRIXLW2X

@misc{pith2026260526263,
  author       = {Pith},
  title        = {Pith review of: On planar functions over $\mathbbF_q^3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRIXLW2X}},
  note         = {Machine review of arXiv:2605.26263}
}
abstract

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 1 canonical work pages

  1. [1]

    Bartoli, M

    D. Bartoli, M. Bonini,Planar polynomials arising from linearized polynomials, Journal of Algebra and Its Applications, vol. 21, no. 1, 2250002, 2022

    2022

  2. [2]

    Bartoli, M

    D. Bartoli, M. Timpanella,A family of planar binomials in characteristic2, Finite Fields and Their Applications, vol. 63, 2020

    2020

  3. [3]

    Blondeau, K

    C. Blondeau, K. Nyberg,Perfect nonlinear functions and cryptography, Finite Fields and Their Applications, vol. 32, pp. 120-147, 2015

    2015

  4. [4]

    Budaghyan, T

    L. Budaghyan, T. Helleseth,New commutative semifields defined by new PN multinomials, Cryptography and Communications, vol. 3, no. 1, pp. 1-16, 2011

    2011

  5. [5]

    Caullery, K.U

    F. Caullery, K.U. Schmidt, Y. Zhou,Exceptional planar polynomials, Designs, Codes and Cryptography, vol. 78, no. 3, pp. 605-613, 2016. ON PLANAR FUNCTIONS OVERF q3 15

    2016

  6. [6]

    C. H. Chan, M. Xiong,Classification of a class of planar quadrinomials, available at arXiv:2404.14291

    work page arXiv

  7. [7]

    R. Chen, S. Mesnager,Characterizations of a class of planar functions over finite fields, Finite Fields and Their Applications, vol. 95, 102382, 2024

    2024

  8. [8]

    Coulter, R.W

    R.S. Coulter, R.W. Matthews,Planar functions and planes of Lenz-Barlotti class II, Designs, Codes and Cryptography, vol. 10, no. 2, pp. 167-184, 1997

    1997

Show all 22 references
  1. [9]

    Dembowski, T

    P. Dembowski, T. G. Ostrom,Planes of ordernwith collineation groups of ordern2, Mathe- matische Zeitschrift, vol. 103, pp. 239-258, 1968

    1968

  2. [10]

    C. Ding, Q. Xiang, J. Yuan, P. Yuan,Explicit classes of permutation polynomials ofF33m, Science in China Series A: Mathematics, vol. 53, no. 4, pp. 639-647, 2009

    2009

  3. [11]

    C. Ding, J. Yin,Signal sets from functions with optimum nonlinearity, IEEE Transactions on Communications, vol. 55, no. 5, pp. 936-940, 2007

    2007

  4. [12]

    Hernando, G

    F. Hernando, G. McGuire, F. Monserrat,On the classification of exceptional planar functions overF p, Geometriae Dedicata, vol. 173, no. 1, pp. 1-35, 2014

    2014

  5. [13]

    Nyberg,Differentially uniform mappings for cryptography, in Advances in Cryptology — EUROCRYPT ’93, Lecture Notes in Computer Science, vol

    K. Nyberg,Differentially uniform mappings for cryptography, in Advances in Cryptology — EUROCRYPT ’93, Lecture Notes in Computer Science, vol. 765, Springer, pp. 55-64, 1994

    1994

  6. [14]

    Nyberg, L

    K. Nyberg, L. R. Knudsen,Provable security against differential cryptanalysis, in Advances in Cryptology — CRYPTO ’92, Lecture Notes in Computer Science, vol. 470, Springer, pp. 566-574, 1993

    1993

  7. [15]

    Kyureghyan, F

    G. Kyureghyan, F. Özbudak,Planarity of products of two linearized polynomials, Finite Fields and Their Applications, vol. 18, no. 6, pp. 1076-1088, 2012

    2012

  8. [16]

    R. Lidl, H. Niederreiter,Finite Fields, Cambridge University Press, 1997

    1997

  9. [17]

    Pott,Almost perfect and planar functions, Designs, Codes and Cryptography, vol

    A. Pott,Almost perfect and planar functions, Designs, Codes and Cryptography, vol. 78, no. 1, pp. 141-195, 2016

    2016

  10. [18]

    Schmidt, Y

    K.-U. Schmidt, Y. Zhou,Planar functions over fields of characteristic two, Journal of Alge- braic Combinatorics, vol. 40, no. 2, pp. 503-526, 2014

    2014

  11. [19]

    Segre, U

    B. Segre, U. Bartocci,Ovali ed altre curve nei piani di Galois di caratteristica due, Acta Arithmetica, vol. 18, pp. 423-449, 1971

    1971

  12. [20]

    J. Yuan, C. Carlet, C. Ding,The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 712-717, 2006

    2006

  13. [21]

    Zhou,(2 n,2 n,2 n,1)-relative difference sets and their representations, Journal of Combina- torial Designs, vol

    Y. Zhou,(2 n,2 n,2 n,1)-relative difference sets and their representations, Journal of Combina- torial Designs, vol. 21, pp. 563-584, 2013

    2013

  14. [22]

    M. E. Zieve,Planar functions and perfect nonlinear monomials over finite fields, Designs, Codes and Cryptography, vol. 75, pp. 71-80, 2015. Centro de Ciências Exatas e Tecnologia - Universidade Federal do Maranhão, A v. dos Portugueses, 1966 - Bacanga, São Luís - MA, Brazil. E...

    2015

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.