# 函数方程f15(z)+g15(z)+h15(z)+w15(z) =1的亚纯函数解的研究The Study of Meromorphic Function Solution of Function Equation of f15(z)+g15(z)+h15(z)+w15(z) =1

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This paper proves that nonconstant meromorphic function in the pole is not more than a single pole with public, and , the solution of Fermat function equations does not exist.

1. 引言及主要结果

${f}^{n}+{g}^{n}+{h}^{n}+{w}^{n}=1$ (1)

2. 几个辅助结果

.

$W\left({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{k}\right)\triangleq |\begin{array}{ccc}{\psi }_{1}& \cdots & {\psi }_{k}\\ {{\psi }^{\prime }}_{1}& \cdots & {{\psi }^{\prime }}_{k}\\ ⋮& \ddots & ⋮\\ {\psi }_{1}^{\left(k-1\right)}& \cdots & {\psi }_{k}^{\left(k-1\right)}\end{array}|\overline{)\equiv }0$ .

${f}^{15}+{g}^{15}+{h}^{15}+{w}^{15}=1$ , (2)

$\tau =|\begin{array}{cccc}{f}^{3}& {g}^{3}& {h}^{3}& {w}^{3}\\ {f}^{2}{f}^{\prime }& {g}^{2}{g}^{\prime }& {h}^{2}{h}^{\prime }& {w}^{2}{w}^{\prime }\\ 14f{\left({f}^{\prime }\right)}^{2}+{f}^{2}{f}^{″}& 14g{\left({g}^{\prime }\right)}^{2}+{g}^{2}{g}^{″}& 14h{\left({h}^{\prime }\right)}^{2}+{h}^{2}{h}^{″}& 14w{\left({w}^{\prime }\right)}^{2}+{w}^{2}{w}^{″}\\ {L}_{3}\left(f\right)& {L}_{3}\left(g\right)& {L}_{3}\left(h\right)& {L}_{3}\left(w\right)\end{array}|$ ,

$\tau$ 是整函数。

$W\left({f}^{15},{g}^{15},{h}^{15},{w}^{15}\right)=|\begin{array}{cccc}{f}^{15}& {g}^{15}& {h}^{15}& {w}^{15}\\ 15{f}^{14}{f}^{\prime }& 15{g}^{14}{g}^{\prime }& 15{h}^{14}{h}^{\prime }& 15{w}^{14}{w}^{\prime }\\ {L}_{1}\left(f\right)& {L}_{1}\left(g\right)& {L}_{1}\left(h\right)& {L}_{1}\left(w\right)\\ {L}_{2}\left(f\right)& {L}_{2}\left(g\right)& {L}_{2}\left(h\right)& {L}_{2}\left(w\right)\end{array}|=3375{f}^{12}{g}^{12}{h}^{12}{w}^{12}\tau \overline{)\equiv }0$ (3)

$\tau \overline{)\equiv }0$ 。此外，结合函数方程 ${f}^{15}+{g}^{15}+{h}^{15}+{w}^{15}=1$ 以及伏朗斯基行列式的特点可得到：

$\begin{array}{l}W\left({f}^{15},{g}^{15},{h}^{15},{w}^{15}\right)=|\begin{array}{ccc}15{g}^{14}{g}^{\prime }& 15{h}^{14}{h}^{\prime }& 15{w}^{14}{w}^{\prime }\\ {L}_{1}\left(g\right)& {L}_{1}\left(h\right)& {L}_{1}\left(w\right)\\ {L}_{2}\left(g\right)& {L}_{2}\left(h\right)& {L}_{2}\left(w\right)\end{array}|\\ =3375{g}^{12}{h}^{12}{w}^{12}|\begin{array}{ccc}{g}^{2}{g}^{\prime }& {h}^{2}{h}^{\prime }& {w}^{2}{w}^{\prime }\\ 14g{\left({g}^{\prime }\right)}^{2}+{g}^{2}{g}^{″}& 14h{\left({h}^{\prime }\right)}^{2}+{h}^{2}{h}^{″}& 14w{\left({w}^{\prime }\right)}^{2}+{w}^{2}{w}^{″}\\ {L}_{3}\left(g\right)& {L}_{3}\left(h\right)& {L}_{3}\left(w\right)\end{array}|\end{array}$ (4)

$\tau =\frac{1}{{f}^{12}}|\begin{array}{ccc}{g}^{2}{g}^{\prime }& {h}^{2}{h}^{\prime }& {w}^{2}{w}^{\prime }\\ 14g{\left({g}^{\prime }\right)}^{2}+{g}^{2}{g}^{″}& 14h{\left({h}^{\prime }\right)}^{2}+{h}^{2}{h}^{″}& 14w{\left({w}^{\prime }\right)}^{2}+{w}^{2}{w}^{″}\\ {L}_{3}\left(g\right)& {L}_{3}\left(h\right)& {L}_{3}\left(w\right)\end{array}|$ (5)

(6)

(7)

$\tau =\frac{1}{{w}^{12}}|\begin{array}{ccc}{f}^{2}{f}^{\prime }& {g}^{2}{g}^{\prime }& {h}^{2}{h}^{\prime }\\ 14f{\left({f}^{\prime }\right)}^{2}+{f}^{2}{f}^{″}& 14g{\left({g}^{\prime }\right)}^{2}+{g}^{2}{g}^{″}& 14h{\left({h}^{\prime }\right)}^{2}+{h}^{2}{h}^{″}\\ {L}_{3}\left(f\right)& {L}_{3}\left(g\right)& {L}_{3}\left(h\right)\end{array}|$ (8)

$f\left(z\right)=\frac{A}{{\left(z-{z}_{\infty }\right)}^{m}}\left(1+o\left(1\right)\right)$ , ,

, $w\left(z\right)=\frac{D}{{\left(z-{z}_{\infty }\right)}^{q}}\left(1+o\left(1\right)\right)$ ,

$\tau =\frac{{g}^{2}{g}^{\prime }{h}^{2}{h}^{\prime }{w}^{2}{w}^{\prime }}{{f}^{12}}|\begin{array}{ccc}1& 1& 1\\ 14\frac{{g}^{\prime }}{g}+\frac{{g}^{″}}{{g}^{\prime }}& 14\frac{{h}^{\prime }}{h}+\frac{{h}^{″}}{{h}^{\prime }}& 14\frac{{w}^{\prime }}{w}+\frac{{w}^{″}}{{w}^{\prime }}\\ {L}_{4}\left(g\right)& {L}_{4}\left(h\right)& {L}_{4}\left(w\right)\end{array}|$ , (9)

$\begin{array}{l}|\begin{array}{ccc}1& 1& 1\\ 14\frac{{g}^{\prime }}{g}+\frac{{g}^{″}}{{g}^{\prime }}& 14\frac{{h}^{\prime }}{h}+\frac{{h}^{″}}{{h}^{\prime }}& 14\frac{{w}^{\prime }}{w}+\frac{{w}^{″}}{{w}^{\prime }}\\ {L}_{4}\left(g\right)& {L}_{4}\left(h\right)& {L}_{4}\left(w\right)\end{array}|\\ =\left\{\left[\left(14\frac{{h}^{\prime }}{h}+\frac{{h}^{″}}{{h}^{\prime }}\right)-\left(14\frac{{g}^{\prime }}{g}+\frac{{g}^{″}}{{g}^{\prime }}\right)\right]\left[\left(182{\left(\frac{{w}^{\prime }}{w}\right)}^{2}+42\frac{{w}^{″}}{w}+\frac{{w}^{‴}}{{w}^{\prime }}\right)-\left(182{\left(\frac{{g}^{\prime }}{g}\right)}^{2}+42\frac{{g}^{″}}{g}+\frac{{g}^{‴}}{{g}^{\prime }}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\left[\left(14\frac{{w}^{\prime }}{w}+\frac{{w}^{″}}{{w}^{\prime }}\right)-\left(14\frac{{g}^{\prime }}{g}+\frac{{g}^{″}}{{g}^{\prime }}\right)\right]\left[\left(182{\left(\frac{{h}^{\prime }}{h}\right)}^{2}+42\frac{{h}^{″}}{h}+\frac{{h}^{‴}}{{h}^{\prime }}\right)-\left(182{\left(\frac{{g}^{\prime }}{g}\right)}^{2}+42\frac{{g}^{″}}{g}+\frac{{g}^{‴}}{{g}^{\prime }}\right)\right]\right\}\end{array}$ (10)

$12m-3\left(n+p+q+1\right)\ge 3$ (*)

I) 若 $m=n=p=q\ge 2$

II) 若

III) 若 $m=n>p\ge q\ge 1$

IV) $m=n\ge p>q\ge 1$

V) $m=n\ge p>q=0$

VI) $m=n=p=q=1$

$g\left(z\right)=\frac{B}{z}+{B}_{0}+{B}_{1}z+{B}_{2}{z}^{2}+{B}_{3}{z}^{3}+O\left({z}^{4}\right)$

$h\left(z\right)=\frac{C}{z}+{C}_{0}+{C}_{1}z+{C}_{2}{z}^{2}+{C}_{3}{z}^{3}+O\left({z}^{4}\right)$$w\left(z\right)=\frac{D}{z}+{D}_{0}+{D}_{1}z+{D}_{2}{z}^{2}+{D}_{3}{z}^{3}+O\left({z}^{4}\right)$

3. 定理1的证明

${\tau }^{4}=\frac{1}{{\left(fghw\right)}^{3}}|\begin{array}{ccc}\frac{{g}^{\prime }}{g}& \frac{{h}^{\prime }}{h}& \frac{{w}^{\prime }}{w}\\ {L}_{5}\left(g\right)& {L}_{5}\left(h\right)& {L}_{5}\left(w\right)\\ {L}_{6}\left(g\right)& {L}_{6}\left(h\right)& {L}_{6}\left(w\right)\end{array}|\cdot |\begin{array}{ccc}\frac{{f}^{\prime }}{f}& \frac{{h}^{\prime }}{h}& \frac{{w}^{\prime }}{w}\\ {L}_{5}\left(f\right)& {L}_{5}\left(h\right)& {L}_{5}\left(w\right)\\ {L}_{6}\left(f\right)& {L}_{6}\left(h\right)& {L}_{6}\left(g\right)\end{array}|\cdot |\begin{array}{ccc}\frac{{f}^{\prime }}{f}& \frac{{g}^{\prime }}{g}& \frac{{w}^{\prime }}{w}\\ {L}_{5}\left(f\right)& {L}_{5}\left(g\right)& {L}_{5}\left(w\right)\\ {L}_{6}\left(f\right)& {L}_{6}\left(g\right)& {L}_{6}\left(w\right)\end{array}|\cdot |\begin{array}{ccc}\frac{{f}^{\prime }}{f}& \frac{{g}^{\prime }}{g}& \frac{{h}^{\prime }}{h}\\ {L}_{5}\left(f\right)& {L}_{5}\left(g\right)& {L}_{5}\left(h\right)\\ {L}_{6}\left(f\right)& {L}_{6}\left(g\right)& {L}_{6}\left(h\right)\end{array}|$

$\begin{array}{c}{\tau }^{15}=\frac{{\tau }^{3}}{{\left(fghw\right)}^{9}}{|\begin{array}{ccc}\frac{{g}^{\prime }}{g}& \frac{{h}^{\prime }}{h}& \frac{{w}^{\prime }}{w}\\ {L}_{5}\left(g\right)& {L}_{5}\left(h\right)& {L}_{5}\left(w\right)\\ {L}_{6}\left(g\right)& {L}_{6}\left(h\right)& {L}_{6}\left(w\right)\end{array}|}^{3}\cdot {|\begin{array}{ccc}\frac{{f}^{\prime }}{f}& \frac{{h}^{\prime }}{h}& \frac{{w}^{\prime }}{w}\\ {L}_{5}\left(f\right)& {L}_{5}\left(h\right)& {L}_{5}\left(w\right)\\ {L}_{6}\left(f\right)& {L}_{6}\left(h\right)& {L}_{6}\left(g\right)\end{array}|}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot {|\begin{array}{ccc}\frac{{f}^{\prime }}{f}& \frac{{g}^{\prime }}{g}& \frac{{w}^{\prime }}{w}\\ {L}_{5}\left(f\right)& {L}_{5}\left(g\right)& {L}_{5}\left(w\right)\\ {L}_{6}\left(f\right)& {L}_{6}\left(g\right)& {L}_{6}\left(w\right)\end{array}|}^{3}\cdot {|\begin{array}{ccc}\frac{{f}^{\prime }}{f}& \frac{{g}^{\prime }}{g}& \frac{{h}^{\prime }}{h}\\ {L}_{5}\left(f\right)& {L}_{5}\left(g\right)& {L}_{5}\left(h\right)\\ {L}_{6}\left(f\right)& {L}_{6}\left(g\right)& {L}_{6}\left(h\right)\end{array}|}^{3}\end{array}$ (**)

${\rho }_{f}={\rho }_{{f}^{\prime }}={\rho }_{{f}^{″}}={\rho }_{{f}^{‴}}$ , ${\rho }_{g}={\rho }_{{g}^{\prime }}={\rho }_{{g}^{″}}={\rho }_{{g}^{‴}}$ ,

${\rho }_{h}={\rho }_{{h}^{\prime }}={\rho }_{{h}^{″}}={\rho }_{{h}^{‴}}$ ,

$F\left(z\right)=f\left({z}^{2}\right),\text{\hspace{0.17em}}G\left(z\right)=g\left({z}^{2}\right),\text{\hspace{0.17em}}H\left(z\right)=h\left({z}^{2}\right),\text{\hspace{0.17em}}W\left(z\right)=w\left(z2\right)$

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[2] Gundersen, G.G. (2003) Complex Functional Equations.

[3] 苏敏, 李玉华. 关于函数方程非平凡亚纯解的研究[J]. 云南师范大学学报: 自然科学版, 2009, 29(2): 44.

[4] 仪洪勋, 杨重骏. 亚纯函数唯一性理论[M]. 北京: 科学出版社, 1995.

[5] Li, Y.H. (2000) Uniqueness Theorems for Meromorphic Functions of Order Less than One. Northeastern Mathematical Journal, 16, 411-416.

[6] 顾永兴, 庞学诚, 方明亮. 正规族理论及其应用[M]. 北京: 科学出版社, 2007.

 [1] 杨乐. 值分布论及其新研究[M]. 北京: 科学出版社, 1982. [2] Gundersen, G.G. (2003) Complex Functional Equations. [3] 苏敏, 李玉华. 关于函数方程非平凡亚纯解的研究[J]. 云南师范大学学报: 自然科学版, 2009, 29(2): 44. [4] 仪洪勋, 杨重骏. 亚纯函数唯一性理论[M]. 北京: 科学出版社, 1995. [5] Li, Y.H. (2000) Uniqueness Theorems for Meromorphic Functions of Order Less than One. Northeastern Mathematical Journal, 16, 411-416. [6] 顾永兴, 庞学诚, 方明亮. 正规族理论及其应用[M]. 北京: 科学出版社, 2007.