A degree bound for planar functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918176553435136 |
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| author | Beierle, Christof Beyne, Tim |
| author_facet | Beierle, Christof Beyne, Tim |
| contents | Using Stickelberger's theorem on Gauss sums, we show that if $F$ is a planar function on a finite field $\mathbb{F}_q$, then for all non-zero functions $G : \mathbb{F}_q \to \mathbb{F}_q$, we have \begin{equation*} d_{\mathsf{alg}}(G \circ F) - d_{\mathsf{alg}}(G) \le \frac{n(p-1)}{2}, \end{equation*} where $q = p^n$ with $p$ a prime and $n$ a positive integer, and $d_{\mathsf{alg}}(F)$ is the algebraic degree of $F$, i.e., the maximum degree of the corresponding system of $n$ lowest-degree interpolating polynomials for $F$ considered as a function on $\mathbb{F}_p^n$. This bound implies the (known) classification of planar polynomials over $\mathbb{F}_p$ and planar monomials over $\mathbb{F}_{p^2}$. As a new result, using the same degree bound, we complete the classification of planar monomials for all $n = \smash{2^k}$ with $p>5$ and $k$ a non-negative integer. Finally, we state a conjecture on the sum of the base-$p$ digits of integers modulo $q-1$ that implies the complete classification of planar monomials over finite fields of characteristic $p>5$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04570 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A degree bound for planar functions Beierle, Christof Beyne, Tim Combinatorics 05B25, 11T06, 11T24 Using Stickelberger's theorem on Gauss sums, we show that if $F$ is a planar function on a finite field $\mathbb{F}_q$, then for all non-zero functions $G : \mathbb{F}_q \to \mathbb{F}_q$, we have \begin{equation*} d_{\mathsf{alg}}(G \circ F) - d_{\mathsf{alg}}(G) \le \frac{n(p-1)}{2}, \end{equation*} where $q = p^n$ with $p$ a prime and $n$ a positive integer, and $d_{\mathsf{alg}}(F)$ is the algebraic degree of $F$, i.e., the maximum degree of the corresponding system of $n$ lowest-degree interpolating polynomials for $F$ considered as a function on $\mathbb{F}_p^n$. This bound implies the (known) classification of planar polynomials over $\mathbb{F}_p$ and planar monomials over $\mathbb{F}_{p^2}$. As a new result, using the same degree bound, we complete the classification of planar monomials for all $n = \smash{2^k}$ with $p>5$ and $k$ a non-negative integer. Finally, we state a conjecture on the sum of the base-$p$ digits of integers modulo $q-1$ that implies the complete classification of planar monomials over finite fields of characteristic $p>5$. |
| title | A degree bound for planar functions |
| topic | Combinatorics 05B25, 11T06, 11T24 |
| url | https://arxiv.org/abs/2407.04570 |