On algebroid solutions of some algebraic differential equations in the complex plane (Q923177)

The author generalizes one of his previous results on binomial differential equations. Let \(a_{jk}\), \(j=0,...,n\); \(k=0,...,q_ j\) be entire functions without common zero for which \(a_{0q_ 0}\cdot a_{nq_ n}\neq 0\). Put \(Q_ j(z,w)=\sum^{q_ j}_{k=0}a_{jk}w^ k,\) \(q_ j=\deg Q_ j\) and consider the differential equation \(\sum^{n}_{j=1}Q_ j(z,w)(w')^ j=Q_ 0(z,w)\) under the conditions \(a_{nq}\) is a polynomial; \(p<q+n\), where \(q=q_ n\) and \(p=\max \{q_ j+j:\) \(0\leq j\leq n-1\}\). Let \(w=w(z)\) be a nonconstant algebroid solution; then it satisfies \[ \min \{n,q+n-p\}\log^+ M(r,w)\leq K\sum_{j,k}\log^+ M(r,a_{jk})+O(\log r), \] for \(r\not\in E\), \(m(E)<\infty\), where \(M(r,v)=\max \{| v(z)|:| z| =r\}\) and K is a constant. If all the \(a_{jk}\) are polynomials, any algebroid solution of the equation is algebraic.