For any positive integers q, n, m with q being a prime power and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish a condition sufficient to ensure the existence of a primitive normal pair <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$(\epsilon ,f(\epsilon ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{n}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mi>n</mi> </msup> </msub> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{PN}_{q^n/q}(\epsilon )=a$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>PN</mtext> <mrow> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo stretchy="false">/</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$a\in \mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is prescribed. Here <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">$f={f_{1}}/{f_{2}}\in \mathbb {F}_{q^n}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mi>n</mi> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a rational function subject to some minor restrictions such that deg(<InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">$f_{1}) + \text {deg}(f_{2})=m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mtext>deg</mtext> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation> and <Equation ID="Equ15"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_Equ15.gif" Format="GIF" Height="70" Rendition="HTML" Resolution="72" Type="Linedraw" Width="220" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \textrm{PN}_{q^n/q}(\epsilon ) =\sum _{i=0}^{n-1}\Bigg (\underset{j\ne i}{\underset{0\le j\le n-1}{\prod }}\epsilon ^{q^j}\Bigg ). \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mtext>PN</mtext> <mrow> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo stretchy="false">/</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo maxsize="2.470em" minsize="2.470em" stretchy="true">(</mo> </mrow> <munder> <munder> <mo>∏</mo> <mrow> <mn>0</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </munder> <mrow> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <msup> <mi>ϵ</mi> <msup> <mi>q</mi> <mi>j</mi> </msup> </msup> <mrow> <mo maxsize="2.470em" minsize="2.470em" stretchy="true">)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Finally, we conclude that for <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$m=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq10.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$q=7^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mn>7</mn> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_814_Article_IEq12.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$k\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, such a pair will exist certainly for all (q, n) except possibly 10 choices at most.

A primitive normal pair with prescribed prenorm

期刊论文
发表日期：

2025年3月25日

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Kaustav Chatterjee,
S K Tiwari

摘要

For any positive integers q, n, m with q being a prime power and $n \ge 5$ $n \geq 5$ , we establish a condition sufficient to ensure the existence of a primitive normal pair $(\epsilon ,f(\epsilon ))$ $(ϵ, f (ϵ))$ in $\mathbb {F}_{q^{n}}$ $F_{q^{n}}$ over $\mathbb {F}_{q}$ $F_{q}$ such that $\textrm{PN}_{q^n/q}(\epsilon )=a$ ${PN}_{q^{n} / q} (ϵ) = a$ , where $a\in \mathbb {F}_{q}$ $a \in F_{q}$ is prescribed. Here $f={f_{1}}/{f_{2}}\in \mathbb {F}_{q^n}(x)$ $f = f_{1} / f_{2} \in F_{q^{n}} (x)$ is a rational function subject to some minor restrictions such that deg( $f_{1}) + \text {deg}(f_{2})=m$ $f_{1}) + deg (f_{2}) = m$ and $\begin{aligned} \textrm{PN}_{q^n/q}(\epsilon ) =\sum _{i=0}^{n-1}\Bigg (\underset{j\ne i}{\underset{0\le j\le n-1}{\prod }}\epsilon ^{q^j}\Bigg ). \end{aligned}$ $\begin{matrix} {PN}_{q^{n} / q} (ϵ) = \sum_{i = 0}^{n - 1} (\underset{j \neq i}{\prod_{0 \leq j \leq n - 1}} ϵ^{q^{j}}) . \end{matrix}$ Finally, we conclude that for $m=3$ $m = 3$ , $n\ge 6$ $n \geq 6$ , and $q=7^k$ $q = 7^{k}$ where $k\in \mathbb {N}$ $k \in N$ , such a pair will exist certainly for all (q, n) except possibly 10 choices at most.