<p>Using the Newton polygon method, we classify a class of generalized Abel equations of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\dot{x} = A(t)x^p + B(t)x^q,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mi>p</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>q</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$2 \le p < q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p, q \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are polynomials with real coefficients. We provide an upper bound for the number of rational solutions of the above equation. An example is presented to show that this bound performs reasonably well.</p>

Rational Solutions of a Class of Generalized Abel Equations

期刊论文
发表日期：

2026年1月16日

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Jiangtao Du,
Yulin Zhao

摘要

Using the Newton polygon method, we classify a class of generalized Abel equations of the form $\dot{x} = A(t)x^p + B(t)x^q,$ $\dot{x} = A (t) x^{p} + B (t) x^{q},$ where $2 \le p < q$ $2 \leq p < q$ , $p, q \in \mathbb {N}$ $p, q \in N$ , and $A(t)$ $A (t)$ , $B(t)$ $B (t)$ are polynomials with real coefficients. We provide an upper bound for the number of rational solutions of the above equation. An example is presented to show that this bound performs reasonably well.