Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t^n + c(at + b)^m \in \mathbb {Z}[t]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>t</mi> <mi>n</mi> </msup> <mo>+</mo> <mi>c</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be any monic irreducible polynomial of degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> with discriminant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$D_{n,m}(a,b,c)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, for any odd integers n and m, we provide a lower bound on the number of tuples of integers (a, b, c) in the interval <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$[D, D + A] \times [E, E + B] \times [F, F + C]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>D</mi> <mo>,</mo> <mi>D</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo>,</mo> <mi>E</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo>,</mo> <mi>F</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$D_{n,m}(a, b, c)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has distinct square-free parts. As a consequence, we obtain a lower bound on the existence of distinct quadratic fields <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{D_{n,m}(a,b,c)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <msub> <mi>D</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We conclude our paper with an improvement of this bound assuming suitable growth conditions on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$A,\ B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mspace width="4pt" /> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> and C.

Distinct square-free parts of discriminants of polynomials and quadratic fields

期刊论文
发表日期：

2026年1月27日

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Anuj Jakhar,
Srinivas Kotyada,
Arunabha Mukhopadhyay

摘要

Let $t^n + c(at + b)^m \in \mathbb {Z}[t]$ $t^{n} + c {(a t + b)}^{m} \in Z [t]$ be any monic irreducible polynomial of degree $n \ge 3$ $n \geq 3$ with discriminant $D_{n,m}(a,b,c)$ $D_{n, m} (a, b, c)$ . In this paper, for any odd integers n and m, we provide a lower bound on the number of tuples of integers (a, b, c) in the interval $[D, D + A] \times [E, E + B] \times [F, F + C]$ $[D, D + A] \times [E, E + B] \times [F, F + C]$ such that $D_{n,m}(a, b, c)$ $D_{n, m} (a, b, c)$ has distinct square-free parts. As a consequence, we obtain a lower bound on the existence of distinct quadratic fields $\mathbb {Q}(\sqrt{D_{n,m}(a,b,c)})$ $Q (\sqrt{D_{n, m} (a, b, c)})$ . We conclude our paper with an improvement of this bound assuming suitable growth conditions on $A,\ B$ $A, B$ and C.