n-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. Constructing n-to-1 mappings has attracted many scholars’ interest. Let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2958_Article_IEq5.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> be the finite field with q elements. In this paper, with the help of AGW-like criteria, by discussing the factorizations of corresponding algebraic curves, we give the complete classification of a class of 3-to-1 quadrinomials of the form <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2958_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="273" /> </InlineMediaObject> <EquationSource Format="TEX">$x^3(x^{3q-3}+bx^{2q-2}+cx^{q-1}+d)\in \mathbb {F}_q[x]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mn>3</mn> <mi>q</mi> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow> <mn>2</mn> <mi>q</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow> <mi>q</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2958_Article_IEq7.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{2}}^*$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>∗</mo> </msubsup> </math></EquationSource> </InlineEquation> in both even and odd characteristic cases.

Complete Classification of 3-to-1 Quadrinomials of the Form $x^3(x^{3q-3}+bx^{2q-2}+cx^{q-1}+d)\in \mathbb {F}_q[x]$ on $\mathbb {F}_{q^2}^*$

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发表日期：

2025年9月26日

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Xiaoer Qin,
Li Yan

摘要

n-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. Constructing n-to-1 mappings has attracted many scholars’ interest. Let $\mathbb {F}_{q}$ $F_{q}$ be the finite field with q elements. In this paper, with the help of AGW-like criteria, by discussing the factorizations of corresponding algebraic curves, we give the complete classification of a class of 3-to-1 quadrinomials of the form $x^3(x^{3q-3}+bx^{2q-2}+cx^{q-1}+d)\in \mathbb {F}_q[x]$ $x^{3} (x^{3 q - 3} + b x^{2 q - 2} + c x^{q - 1} + d) \in F_{q} [x]$ over $\mathbb {F}_{q^{2}}^*$ $F_{q^{2}}^{*}$ in both even and odd characteristic cases.

Complete Classification of 3-to-1 Quadrinomials of the Form \(x^3(x^{3q-3}+bx^{2q-2}+cx^{q-1}+d)\in \mathbb {F}_q[x]\) on \(\mathbb {F}_{q^2}^*\)

摘要