一类Schwarzian微分方程的亚纯解

The Meromorphic Solutions of a Certain Type of the Schwarzian Differential Equation

研究了如下形式的Schwarzian微分方程亚纯解的存在与结构问题:(f″/f')'-1/2(f″/f')=Pm(z)/Qn(z),其中Pm(z),Qn(z)是次数分别为m,n的不可约多项式,满足m≤n-2.已有研究结果给出了在该方程存在亚纯解时Pm(z)/Qn(z)的特征,涉及比较复杂的行列式运算.该文找到了替换行列式运算的新检验条件,通过判断有限个方程组是否有解来判断Schwarzian方程是否存在亚纯解,并给出了亚纯解的具体结构.

In this paper,meromorphic solutions of the following type of the Schwarzian differential equation f″f′′-12f″f′2=P m(z)Q n(z)are studied.Here P m(z),Q n(z)are two irreducible polynomials of degree m,n satisfying m≤n-2.The characterization of P m(z)Q n(z)for ensuring the existence of meromorphic solutions is known,but very complicated in the calculation of determinants.A new condition,which tests the existence of solutions of a series of equation sets,has been found in this paper,to replace the determinant requirements.Furthermore,the solution structures are also given out for the first time.