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<p>It is shown that, out of a large class of delay partial differential equations, there is only a narrow class of equations, containing the Kac–Van Moerbeke equation, that can admit meromorphic solutions <i>w</i> satisfying the growth condition</p><p><Equation ID="Equ1"> <EquationNumber>(1)</EquationNumber> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2756_Article_Equ1.gif" Format="GIF" Height="40" Rendition="HTML" Resolution="72" Type="Linedraw" Width="165" /> </MediaObject> <EquationSource Format="TEX">\(\mathop {\lim \sup}\limits_{r \to \infty} {{\log T({r,w})} \over r} = 0,\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mrow> <mo form="prefix" movablelimits="true">lim</mo> <mo form="prefix" movablelimits="true">sup</mo> </mrow> <mrow> <mi>r</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mfrac> <mrow> <mi>log</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mrow> <mi>r</mi> <mo>,</mo> <mi>w</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </math></EquationSource> </Equation></p><p>where <i>T</i>(<i>r, w</i>) is the Nevanlinna characteristic of <i>w</i>. Condition (<InternalRef RefID="Equ1">1</InternalRef>) is, in the sense of Vojta’s dictionary, a direct analogue of the zero algebraic entropy criterion introduced by Hietarinta and Viallet.</p>

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Growth of meromorphic solutions and delay partial differential equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月9日

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  • Tingbin Cao,
  • Risto Korhonen

摘要

It is shown that, out of a large class of delay partial differential equations, there is only a narrow class of equations, containing the Kac–Van Moerbeke equation, that can admit meromorphic solutions w satisfying the growth condition

(1) \(\mathop {\lim \sup}\limits_{r \to \infty} {{\log T({r,w})} \over r} = 0,\) lim sup r → ∞ log T ( r , w ) r = 0 ,

where T(r, w) is the Nevanlinna characteristic of w. Condition (1) is, in the sense of Vojta’s dictionary, a direct analogue of the zero algebraic entropy criterion introduced by Hietarinta and Viallet.