Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> denote the ring of integers of the number field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K = \mathbb {Q}(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> is a root of the monic irreducible polynomial <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(x) \in \mathbb {Z}[x]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. We say that f(x) is monogenic if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Z}_K = \mathbb {Z}[\theta ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">Z</mi> <mi>K</mi> </msub> <mo>=</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <mi>θ</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. A polynomial <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f(x) \in \mathbb {Z}[x]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is called reciprocal if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f(x) = x^{\operatorname {deg}(f)} f(1/x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow> <mo>deg</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this article, we derive sufficient conditions for the monogeneity of even degree reciprocal polynomials. By employing properties of the discriminant of reciprocal polynomials, we partially prove a conjecture proposed by Jones in 2021. Furthermore, we establish a lower bound on the number of certain sextic monogenic reciprocal polynomials.

On monogeneity of reciprocal polynomials

期刊论文
发表日期：

2026年2月12日

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Rupam Barman,
Anuj Narode,
Vinay Wagh

摘要

Let $\mathbb {Z}_K$ $Z_{K}$ denote the ring of integers of the number field $K = \mathbb {Q}(\theta )$ $K = Q (θ)$ , where $\theta $ $θ$ is a root of the monic irreducible polynomial $f(x) \in \mathbb {Z}[x]$ $f (x) \in Z [x]$ . We say that f(x) is monogenic if $\mathbb {Z}_K = \mathbb {Z}[\theta ]$ $Z_{K} = Z [θ]$ . A polynomial $f(x) \in \mathbb {Z}[x]$ $f (x) \in Z [x]$ is called reciprocal if $f(x) = x^{\operatorname {deg}(f)} f(1/x)$ $f (x) = x^{deg (f)} f (1 / x)$ . In this article, we derive sufficient conditions for the monogeneity of even degree reciprocal polynomials. By employing properties of the discriminant of reciprocal polynomials, we partially prove a conjecture proposed by Jones in 2021. Furthermore, we establish a lower bound on the number of certain sextic monogenic reciprocal polynomials.