On planar functions over $\mathbb{F}_{q^3}$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918523023917056 |
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| author | Guardieiro, João Paulo Marques, Adler Quoos, Luciane Tizziotti, Guilherme |
| author_facet | Guardieiro, João Paulo Marques, Adler Quoos, Luciane Tizziotti, Guilherme |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26263 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On planar functions over $\mathbb{F}_{q^3}$ Guardieiro, João Paulo Marques, Adler Quoos, Luciane Tizziotti, Guilherme Number Theory Combinatorics 11T06, 11C20, 12E10 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. |
| title | On planar functions over $\mathbb{F}_{q^3}$ |
| topic | Number Theory Combinatorics 11T06, 11C20, 12E10 |
| url | https://arxiv.org/abs/2605.26263 |