A monic polynomial <InlineEquation ID="IEq1"> <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> of degree n is called monogenic if f(x) is irreducible over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{1,\theta ,\theta ^2,\ldots ,\theta ^{n-1}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <msup> <mi>θ</mi> <mn>2</mn> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>θ</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is a basis for the ring of integers of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb Q}(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f(\theta )=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. A strictly-Perron polynomial is the minimal polynomial of a Perron number <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is neither a Pisot number, an anti-Pisot number, nor a Salem number. For any natural number <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove that there exist infinitely many monogenic strictly-Perron polynomials of degree n.

Monogenic strictly-Perron polynomials

期刊论文
发表日期：

2026年6月24日

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Lenny Jones

摘要

A monic polynomial $f(x)\in {\mathbb Z}[x]$ $f (x) \in Z [x]$ of degree n is called monogenic if f(x) is irreducible over ${\mathbb Q}$ $Q$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{n-1}\}$ ${1, θ, θ^{2}, \dots, θ^{n - 1}}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$ $Q (θ)$ , where $f(\theta )=0$ $f (θ) = 0$ . A strictly-Perron polynomial is the minimal polynomial of a Perron number $\lambda $ $λ$ such that $\lambda $ $λ$ is neither a Pisot number, an anti-Pisot number, nor a Salem number. For any natural number $n\ge 2$ $n \geq 2$ , we prove that there exist infinitely many monogenic strictly-Perron polynomials of degree n.