Several classes of permutation polynomials of the form $ (x^{p^m}-x+\delta)^s+L(x) $

Changhui Chen; Haibin Kan; Yanjun Li; Jie Peng; Lijing Zheng

doi:10.3934/amc.2023062

Article Contents

2025, Volume 19, Issue 1: 319-336. Doi: 10.3934/amc.2023062

This issue Previous Article Fractional decoding and the Rosenbloom-Tsfasman metric Next Article Polynomial hashing over prime order fields

Several classes of permutation polynomials of the form $ (x^{p^m}-x+\delta)^s+L(x) $

1.
Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China
2.
Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China
3.
Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
4.
School of Computer Science, Fudan University, Shanghai 200433, China
5.
Shanghai Engineering Research Center of Blockchain, Shanghai 200433, China
6.
Yiwu Research Institute of Fudan University, Yiwu City 322000, China
7.
School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

^*Corresponding author: Yanjun Li
^*Corresponding author: Yanjun Li

Received: May 2023

Revised: November 2023

Early access: January 2, 2024

Published online: January 2, 2024

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Abstract

In this paper, we give six classes of permutation polynomials of the form $ (x^{2^m}+x+\delta)^s+ax $ over $ \mathbb{F}_{2^{2m}} $ and six classes of permutation polynomials of the form $ (x^{p^m}-x+\delta)^s+ax^{p^m}+a^{p^m}x $ over $ \mathbb{F}_{p^{2m}} $ ($ p $ being an odd prime), respectively. In addition, we also investigate permutation polynomials obtained from piecewise functions. Consequently, we find some complete permutation polynomials.

Keywords:

Mathematics Subject Classification: Primary: 05A05, 11T06; Secondary: 11T55.

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Table 1. Known permutation polynomials $(x^{2^m}+x+\delta)^s+L(x)$ over $\mathbb{F}_{2^{n}}$

s	L(x)	conditions	reference
$ s(2^m-1)\equiv0 $ ($ \bmod $ $ 2^{n}-1 $)	$ ax $	all $ \delta $ and $ a\in\mathbb{F}_{2^m}^\ast $	[19]
$ s=i(2^m-1)+1 $	$ a_1x^{2^m}+a_2x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $, $ a_1+a_2\ne0 $, and $ \operatorname{Tr}_m^{n}(a_1+a_2)+{a_2}^{2^m+1} $ +$ {a_1}^{2^m+1}\ne0 $	[19]
$ s=2^{n-1}+2^{m-1} $	$ a_1x^{2^m}+a_2x $	$ \operatorname{Tr}_m^{n}(\delta)\ne0 $, either $ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, $ a_1^{2^m}+a_2\ne0 $ or $ a_1+a_2\notin\mathbb{F}_{2^m} $, $ \operatorname{Tr}_m^{n}(a_1+a_2)={a_2}^{2^m+1}+{a_1}^{2^m+1} $	[19]
$ s\in\{i(2^m-1)+1 $, $ (i\cdot2^{m}+1)(2^m-1)+1 $ with $ (2^k+1)i\equiv 1 $ $ (\bmod $ $ 2^m+1) $	$ a_1x^{2^m}+a_2x $	$ \operatorname{Tr}_m^{n}(\delta)\ne0 $, $ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, and $ \frac{a_1^{2^m+1}+a_2^{2^m+1}}{a_1+a_2}\in\mathbb{F}_{2^{\text{gcd}(k,m)}}^\ast $	[19]
$ s\in\{i(2^m-1)+1 $, $ (i\cdot2^{m}+1)(2^m-1)+1 $ with $ (2^k-1)i\equiv -1 $ $ (\bmod $ $ 2^m+1) $	$ a_1x^{2^m}+a_2x $	$ \operatorname{Tr}_m^{n}(\delta)\ne0 $, $ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, and $ \frac{a_1^{2^m+1}+a_2^{2^m+1}}{a_1+a_2}\in\mathbb{F}_{2^{\text{gcd}(k,m)}}^\ast $	[19]
$ s\in\{2^{m+1}+3,3\cdot2^m+2\} $	$ a_1x^{2^m}+a_2x $	$ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, $ a_1^{2^m}\ne a_2 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \operatorname{Tr}_1^{m}(\frac{\operatorname{Tr}_m^{n}(\delta^4)}{a_1^{2^m}+a_2}+1)=0 $ and $ p_m(\frac{a_1^{2^m}+a_2}{\operatorname{Tr}_m^{n}(\delta^4)})\ne0 $	[19]
$ s\in\{3\cdot2^{n-2}+2^{m-2} $, $ 2^{n-2}+3\cdot2^{m-2}\} $}	$ a_1x^{2^m}+a_2x $	$ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, $ a_1^{2^m}\ne a_2 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \operatorname{Tr}_1^{m}((\frac{1}{a_1^{2^m}+a_2})^2+1)=0 $ and $ p_m((a_1^{2^m}+a_2)^2)\ne0 $	[19]
$ s\in\{2^{n-2}+2^{m-2}+1 $, $ 2^{n-2}+2^{m-2}+2^m\} $	$ a_1x^{2^m}+a_2x $	$ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, $ a_1^{2^m}\ne a_2 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \operatorname{Tr}_1^{m}(\frac{\operatorname{Tr}_m^{n}(\delta)}{a_1^{2^m}+a_2}+1)=0 $ and $ p_m(\frac{(a_1^{2^m}+a_2)^2}{\operatorname{Tr}_m^{n}(\delta)})\ne0 $	[19]
$ s\in\{2^{n-1}+2^{m+1}+2 $, $ 5\cdot2^{m-1}+2\} $}	$ a_1x^{2^m}+a_2x $	$ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, $ a_1^{2^m}\ne a_2 $ either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \operatorname{Tr}_1^{m}(\frac{(\operatorname{Tr}_m^{n}(\delta))^\frac{7}{2}}{a_1^{2^m}+a_2}+1)=0 $ and $ p_m(\frac{(a_1^{2^m}+a_2)^2}{(\operatorname{Tr}_m^{n}(\delta))^\frac{7}{2}})\ne0 $	[19]
$ s\in\{2^{n-2}+3\cdot2^{m-3}+2 $, $ 3\cdot2^{n-3}+2^{m-2}\} $	$ a_1x^{2^m}+a_2x $	$ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, $ a_1^{2^m}\ne a_2 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \operatorname{Tr}_1^{m}(\frac{1}{(a_1^{2^m}+a_2)^2\cdot(\operatorname{Tr}_m^{n}(\delta))^\frac{3}{4}}+1)=0 $ and $ p_m((a_1^{2^m}+a_2)^2\cdot(\operatorname{Tr}_m^{n}(\delta))^\frac{3}{4})\ne0 $	[19]
$ s\in\{2^{n-2}+5\cdot2^{n-4} $, $ 5\cdot2^{m-4}+2^{n-2}\} $	$ a_1x^{2^m}+a_2x $	$ a_1+a_2\in\mathbb{F}_{2^m}^\ast\backslash\{1\} $, $ a_1^{2^m}\ne a_2 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \operatorname{Tr}_1^{m}(\frac{1}{(a_1^{2^m}+a_2)^2\cdot(\operatorname{Tr}_m^{n}(\delta))^\frac{7}{8}}+1)=0 $ and $ p_m((a_1^{2^m}+a_2)^2\cdot(\operatorname{Tr}_m^{n}(\delta))^\frac{7}{8})\ne0 $	[19]
$ s $ is even	$ x $	$ \operatorname{Tr}_m^{2m}(\delta)=0 $	[21]
$ s=3\cdot2^{n-2}+2^{m-2} $	$ x $	all $ \delta $ and $ m\not\equiv0 $ $ (\bmod $ 3)	[29]
$ s=\frac{2^{n}+2^m+1}{3} $	$ x $	all $ \delta $ and $ m $ is even	[29]
$ s=2^{n-2}+2^{m-2}+1 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \operatorname{Tr}_1^{n}(\delta)=\operatorname{Tr}_1^m(1) $ and $ p_m(\frac{1}{\operatorname{Tr}_m^{n}(\delta)})\ne0 $	[29]
$ s=2^{m+1}+1 $	$ a_1x^{2^m}+a_2x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $, $ a_1+a_2\in\mathbb{F}_{2^m}^\ast $, and $ a_1^{2^m}+a_2\notin\{0,\operatorname{Tr}_1^{n}(\delta^2)\} $	[5]
$ s=2^n-2 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $ or 1	[18]
$ s=2^n-3 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)^3=1 $ when $ m $ is even; $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)=1 $ when $ m $ is odd	[18]
$ s=2^i $	$ ax $	all $ \delta $ and $ a\in\mathbb{F}_{2^m}^\ast $	[17]
$ s=2^i+1 $	$ ax $	$ a\in\mathbb{F}_{2^m}^\ast $, $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)^{2^j}=a $ or $ \operatorname{Tr}_m^{n}(\delta)(\frac{a\operatorname{Tr}_m^{n}(\delta)^{2^j}+1}{a})\ne0 $ and $ (\frac{a\operatorname{Tr}_m^{n}(\delta)^{2^j}+1}{\operatorname{Tr}_m^{n}(\delta)})^{\frac{2^m-1}{2^d-1}}\ne1 $, where $ d= $gcd$ (i,m) $, $ j < m $ and $ j\equiv i (\bmod m) $	[17]
$ s=2^i+2^m $	$ ax $	$ a\in\mathbb{F}_{2^m}^\ast $, $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)^{2^j}=a $ or $ \operatorname{Tr}_m^{n}(\delta)(\frac{a\operatorname{Tr}_m^{n}(\delta)^{2^j}+1}{a})\ne0 $ and $ (\frac{a\operatorname{Tr}_m^{n}(\delta)^{2^j}+1}{\operatorname{Tr}_m^{n}(\delta)})^{\frac{2^m-1}{2^d-1}}\ne1 $, where $ d= $gcd$ (i,m) $, $ j < m $ and $ j\equiv i (\bmod m) $	[17]
$ s=2^i+2^m+1 $	$ ax $	$ a\in\mathbb{F}_{2^m}^\ast $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)^{2^i+1}=a $	[17]
$ s=2^{i}-2^{m+1}-2 $	$ x $	$ i > 0 $, $ \operatorname{Tr}_m^{n}(\delta^{2^i})=0 $ or $ 1 $	[17]
$ s=6 $	$ ax $	$ a\in\mathbb{F}_{2^m}^\ast $, $ 2^m>4 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\not=0 $, $ \operatorname{Tr}_1^{m}(\frac{\operatorname{Tr}_m^{n}(\delta)^5+a}{a})=0 $ and $ p_m(\frac{a}{\operatorname{Tr}_m^{n}(\delta)^5})\ne0 $	[17]
$ s=\frac{2^m(2^{m+1}+1)}{4} $	$ ax $	$ a\in\mathbb{F}_{2^m}^\ast $, $ 2^m>4 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\not=0 $, $ \operatorname{Tr}_1^{m}(\frac{a^2\operatorname{Tr}_m^{n}(\delta)^{2^{m-1}}+1}{a^2\operatorname{Tr}_m^{n}(\delta)^{2^{m-1}}})=0 $ and $ p_m(a^2\operatorname{Tr}_m^{n}(\delta)^{2^{m-1}})\ne0 $	[17]
$ s=\frac{2^m(2^{m+1}+3)}{4} $	$ ax $	$ a\in\mathbb{F}_{2^m}^\ast $, $ 2^m>4 $, either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)=a^4 $	[17]
$ s=2^{i}+1 $	$ ax $	$ \operatorname{Tr}_m^{n}(\delta^{2^i})=0 $ or $ a $, $ a\in\mathbb{F}_{2^m}^\ast $ and $ 0 < i < n $ and gcd$ (i,m)=1 $	Theorem 3.1
$ s=2^{m+1}+2^i+2 $	$ ax $	$ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ a\in\mathbb{F}_{2^m}^\ast $, $ \operatorname{Tr}_1^{m}(\frac{(\operatorname{Tr}_m^{n}(\delta))^{2^i+3}}{a}+1)=0 $, and $ p_m(\frac{a}{(\operatorname{Tr}_m^{n}(\delta))^{2^i+3}})\ne0 $	[17] and Theorem 3.3
$ s=3\cdot2^{m}-1 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)=1 $	Theorem 3.5
$ s=2^{m+2}-2 $	$ ax $	$ \operatorname{Tr}_m^{n}(\delta)=a $ and $ a\in\mathbb{F}_{2^m}^\ast $	Theorem 3.6
$ s=i(2^n-2^m-2) $	$ ax $	all $ \delta $ and $ a\in\mathbb{F}_{2^m}^\ast $	Theorem 3.7
$ s=2^{n}-2^{m}-3 $	$ ax $	$ a\operatorname{Tr}_m^{n}(\delta)^4=1 $ and $ a\in\mathbb{F}_{2^m}^\ast $	Theorem 3.8

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Table 2. Known permutation polynomials $(x^{p^m}-x+\delta)^s+L(x)$ over $\mathbb{F}_{p^{n}}$with odd prime p

s	L(x)	conditions	reference
$ s(p^m-1)\equiv0 $ ($ \bmod $ $ p^{n}-1 $)	$ ax,a\in\mathbb{F}_{p^m}^\ast $	all $ \delta $	[19]
$ s=i(p^m-1)+1 $	$ a_1x^{p^m}+a_2x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $, $ a_1+a_2\ne0 $, and $ (-1)^{i+1}\operatorname{Tr}_m^{n}(a_1+a_2)+{a_2}^{p^m+1} $ -$ {a_1}^{p^m+1}\ne0 $	[19]
$ s=p^m+2 $	$ a_1x^{p^m}+a_2x $	$ p=3 $, $ a_1+a_2\ne0 $, and $ (\operatorname{Tr}_m^{n}(\delta))^2+{a_2}-{a_1}^{p^m} $ is a square in $ \mathbb{F}_{p^m} $	[19]
$ s=i(p^m+1)+1 $	$ x^{p^m}+x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $	[15]
$ s=2\cdot p^m+1 $	$ a_1x^{p^m}+a_2x $	$ p=3 $, $ a_1+a_2\ne0 $, and $ (\operatorname{Tr}_m^{n}(\delta))^2-{a_2}+{a_1}^{p^m} $ is a square in $ \mathbb{F}_{p^m} $	[19]
$ s=2\cdot p^{n-1}+p^{m-1} $	$ a_1x^{p^m}+a_2x $	$ p=3 $, $ a_1+a_2\ne0 $, $ a_1^{p^m}\ne {a_2} $, and $ \operatorname{Tr}_m^{n}(\delta)=0 $	[19]
$ s=2\cdot p^{n-1}+p^{m-1} $	$ a_1x^{p^m}+a_2x $	$ p=3 $, $ a_1+a_2\ne0 $, $ a_1^{p^m}- a_2 $ is a square in $ \mathbb{F}_{p^m} $, and $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	[19]
$ s=2\cdot p^{n-1}+p^{m-1} $	$ x $	$ p=3 $ and $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	[29]
$ s $ is even	$ x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $	[21]
$ s=\frac{p^{n}+1}{2} $	$ x^{p^m}+x $	all $ \delta $	[21]
$ s=p^n-2 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)=\pm1 $	[18]
$ s=p^n-p^m-1 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)^2=p-1 $, where $ p-1 $ is a square in $ \mathbb{F}_{p^m} $	[18]
$ s=p^i(p^m+1)+p^m $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)^2=p-2 $ and gcd$ (p^{m-i}+2,p^m-1)=1 $, where $ p-2 $ is a square in $ \mathbb{F}_{p^m} $ and $ 1\le i\le m-1 $	[18]
$ s=p^i(p^m+1)+1 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)^2=2 $ and gcd$ (p^{m-i}+2,p^m-1)=1 $, where $ 2 $ is a square in $ \mathbb{F}_{p^m} $ and $ 1\le i\le m-1 $	[18]
$ s=p^m+p $	$ x $	either $ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \frac{\operatorname{Tr}_m^{n}(\delta)+1}{\operatorname{Tr}_m^{n}(\delta)} $ is a $ (p-1) $-th power in $ \mathbb{F}_{p^m} $	[18]
$ s=2\cdot3^{m}+1 $	$ x $	either $ {\operatorname{Tr}_m^{n}(\delta)^{2}}=1 $ or $ \operatorname{Tr}_m^{n}(\delta)^2-1 $ is a square of $ \mathbb{F}_{3^n} $	[18]
$ s=p^{m+1}+1 $	$ ax^{p^m}+a^{p^m}x $	$ a+a^{p^m}\ne0 $ and $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	[5]
$ s=p^{m+1}+1 $	$ x $	$ \operatorname{Tr}_m^{n}(\delta)=0 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ with $ \frac{\operatorname{Tr}_m^n(\delta)-1}{\operatorname{Tr}_m^n(\delta)} $ is not a $ (p-1) $-th power in $ \mathbb{F}_{p^{n}} $	[5]
$ s=2p^{m} $	$ x^{p^m}+x $	$ \operatorname{Tr}_m^{n}(\delta)\ne0 $	[5]
$ s=p^i+p^m $	$ ax $	$ a\in\mathbb{F}_{p^m}^\ast $, $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ or $ \operatorname{Tr}_m^{n}(\delta)^{p^j}=-a $ or $ \operatorname{Tr}_m^{n}(\delta)(\frac{\operatorname{Tr}_m^{n}(\delta)^{p^j}+b}{b})\ne0 $ and $ (z^{\frac{(p^i-1)(p^m+1)}{2}}\cdot\frac{\operatorname{Tr}_m^{n}(\delta)^{p^j}+a}{\operatorname{Tr}_m^{n}(\delta)})^{\frac{p^n-1}{p^d-1}}\ne1 $, where $ d= $gcd$ (i,m) $, $ j < m $ and $ j\equiv i (\bmod m) $	[17]
$ s=2 $	$ ax $	$ a\in\mathbb{F}_{p^m}^\ast $ and $ \frac{2\operatorname{Tr}_m^{n}(\delta)-a}{a}\ne0 $	[17]
$ s=p^i+1 $	$ ax $	$ a\in\mathbb{F}_{p^m}^\ast $, $ \operatorname{Tr}_m^{n}(\delta)\ne0 $ or $ \operatorname{Tr}_m^{n}(\delta)^{p^j}=-a $ or $ \operatorname{Tr}_m^{n}(\delta)(\frac{\operatorname{Tr}_m^{n}(\delta)^{p^j}+b}{b})\ne0 $ and $ (z^{\frac{(p^i-1)(p^m+1)}{2}}\cdot\frac{\operatorname{Tr}_m^{n}(\delta)^{p^j}+a}{\operatorname{Tr}_m^{n}(\delta)})^{\frac{p^n-1}{p^d-1}}\ne1 $, where $ d= $gcd$ (i,m) $, $ j < m $ and $ j\equiv i (\bmod m) $	[17]
$ s=2p^{i} $	$ ax $	$ 0 < i\le m-1 $, $ a\in\mathbb{F}_{p^m}^\ast $, either $ (\frac{az^{\frac{(p^i-1)(p^m+1)}{2}}}{\operatorname{Tr}_m^{n}(\delta)^{p^i}})^{\frac{p^n-1}{p^d-1}}\ne1 $ or $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	[17]
$ s=2p^{i} $	$ ax^{p^m}+a^{p^m}x $	$ 0 < i < n $, $ a+a^{p^m}\ne0 $, and $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	Theorem 3.11
$ s=\frac{p^n+3}{2} $	$ ax^{p^m}+a^{p^m}x $	$ a+a^{p^m}\ne0 $ and $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	Theorem 3.13
$ s=p^i+1 $	$ ax^{p^m}+a^{p^m}x $	$ 0 < i < n $, gcd$ (2(p^i-1),p^n-1)=4 $, $ p^m-1\equiv0 $ $ (\bmod $ 4), $ a+a^{p^m}\ne0 $, and $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	Theorem 3.14
$ s>0 $ with gcd$ (s,p^m-1)=1 $	$ ax^{p^m}+a^{p^m}x $	$ a+a^{p^m}\ne0 $ and $ \operatorname{Tr}_m^{n}(\delta)=0 $; or all $ \delta\in\mathbb{F}_{p^n} $ if $ s $ is a power of $ p $	Theorem 3.17
$ s=p^m+2 $	$ ax^{p^m}+a^{p^m}x $	$ p=3 $, $ a+a^{p^m}\ne0 $ and all $ \delta $	Theorem 3.18
$ s=i(p^m+1)+1 $	$ ax^{p^m}+a^{p^m}x $	$ a+a^{p^m}\ne0 $ and $ \operatorname{Tr}_m^{n}(\delta)=0 $	Theorem 3.18
$ s=p^m+p^i $	$ ax^{p^m}+a^{p^m}x $	$ i>0 $ with $ i \| m $, $ a+a^{p^m}\ne0 $, and $ \operatorname{Tr}_m^{n}(\delta)\ne0 $	Theorem 3.15

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