This paper is concerned with the non-trivial rational solutions for the generalized Abel equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ x^{\prime }=A(t)x^n+B(t)x^m+C(t)x, $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mi>n</mi> </msup> <mo>+</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mi>m</mi> </msup> <mo>+</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m,n\in \mathbb {N},\,n>m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>n</mi> <mo>></mo> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A(t),B(t),C(t)\in \mathbb {R}[t]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. A solution is called a non-trivial rational solution if it takes the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x=q(t)/p(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where p, q are polynomials in t and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gcd (p,q)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\deg (p)\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>deg</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this work, we provide an upper bound on the number of the real and complex non-trivial rational solutions for the mentioned generalized Abel equation.

An Upper Bound for the Number of Rational Solutions of Generalized Abel Equations

期刊论文
发表日期：

2026年5月9日

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Hui Zeng,
Yulin Zhao

摘要

This paper is concerned with the non-trivial rational solutions for the generalized Abel equation $ x^{\prime }=A(t)x^n+B(t)x^m+C(t)x, $ $x^{'} = A (t) x^{n} + B (t) x^{m} + C (t) x,$ where $m,n\in \mathbb {N},\,n>m>1$ $m, n \in N, n > m > 1$ and $A(t),B(t),C(t)\in \mathbb {R}[t]$ $A (t), B (t), C (t) \in R [t]$ . A solution is called a non-trivial rational solution if it takes the form $x=q(t)/p(t)$ $x = q (t) / p (t)$ , where p, q are polynomials in t and $\gcd (p,q)=1$ $gcd (p, q) = 1$ with $\deg (p)\ge 1$ $deg (p) \geq 1$ . In this work, we provide an upper bound on the number of the real and complex non-trivial rational solutions for the mentioned generalized Abel equation.