<p>In this paper, we investigate the distinct squarefree parts of the discriminants of polynomials of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">$ f(t) = t^n + c(at^k + b)^m $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>t</mi> <mi>n</mi> </msup> <mo>+</mo> <mi>c</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mi>t</mi> <mi>k</mi> </msup> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$ a $</EquationSource> <EquationSource Format="MATHML"><math> <mi>a</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$ b $</EquationSource> <EquationSource Format="MATHML"><math> <mi>b</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$ c $</EquationSource> <EquationSource Format="MATHML"><math> <mi>c</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ k $</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$ m $</EquationSource> <EquationSource Format="MATHML"><math> <mi>m</mi> </math></EquationSource> </InlineEquation> are fixed integers with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$n>km$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mi>k</mi> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$ n $</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> varies over the range <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$ [1, N] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> for an arbitrary positive integer <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2448_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$ N $</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation>. As a consequence, a lower bound on the existence of certain distinct quadratic fields is established.</p>

On Squarefree Parts of Polynomial Discriminants and Quadratic Fields

期刊论文
发表日期：

2025年6月12日

查看全文

Anuj Jakhar,
Srinivas Kotyada,
Arunabha Mukhopadhyay

摘要

In this paper, we investigate the distinct squarefree parts of the discriminants of polynomials of the form $ f(t) = t^n + c(at^k + b)^m $ $f (t) = t^{n} + c {(a t^{k} + b)}^{m}$ , where $ a $ $a$ , $ b $ $b$ , $ c $ $c$ , $ k $ $k$ and $ m $ $m$ are fixed integers with $n>km$ $n > k m$ , and $ n $ $n$ varies over the range $ [1, N] $ $[1, N]$ for an arbitrary positive integer $ N $ $N$ . As a consequence, a lower bound on the existence of certain distinct quadratic fields is established.