<p>Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3717_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="236" /> </InlineMediaObject> <EquationSource Format="TEX">$f(t_1, \ldots , t_r, X)\in {\mathbb {Z}}[t_1, \ldots , t_r,X]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be irreducible and let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3717_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">$a_1, \ldots , a_r\in {\mathbb {Z}}{\setminus } \{0,\pm 1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>r</mi> </msub> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under a necessary ramification assumption on <i>f</i>, and conditionally on the Generalized Riemann Hypothesis, we show that for almost all integers <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3717_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$n_1, \ldots , n_r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, the polynomial <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3717_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">$f(a_1^{n_1}, \ldots , a_r^{n_r}, X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msubsup> <mi>a</mi> <mn>1</mn> <msub> <mi>n</mi> <mn>1</mn> </msub> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>a</mi> <mi>r</mi> <msub> <mi>n</mi> <mi>r</mi> </msub> </msubsup> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is irreducible in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3717_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {Q}}[X]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the irreducibility of $f(2^n,3^m,X)$ and other such polynomials

期刊论文
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发表日期：

2025年3月18日

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Lior Bary-Soroker,
Daniele Garzoni,
Vlad Matei

摘要

Let $f(t_1, \ldots , t_r, X)\in {\mathbb {Z}}[t_1, \ldots , t_r,X]$ $f (t_{1}, \dots, t_{r}, X) \in Z [t_{1}, \dots, t_{r}, X]$ be irreducible and let $a_1, \ldots , a_r\in {\mathbb {Z}}{\setminus } \{0,\pm 1\}$ $a_{1}, \dots, a_{r} \in Z \ {0, \pm 1}$ . Under a necessary ramification assumption on f, and conditionally on the Generalized Riemann Hypothesis, we show that for almost all integers $n_1, \ldots , n_r$ $n_{1}, \dots, n_{r}$ , the polynomial $f(a_1^{n_1}, \ldots , a_r^{n_r}, X)$ $f (a_{1}^{n_{1}}, \dots, a_{r}^{n_{r}}, X)$ is irreducible in ${\mathbb {Q}}[X]$ $Q [X]$ .

On the irreducibility of \(f(2^n,3^m,X)\) and other such polynomials

摘要