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Rational solutions to algebraic differential equations. (English. Russian original) Zbl 0809.34008
Differ. Equations 29, No. 10, 1454-1460 (1993); translation from Differ. Uravn. 29, No. 10, 1675-1683 (1993).
Let us consider the differential equation in the complex plane $\sum^ N_{i=0} B_ i(z) \prod^ n_{k=0} \{w^{(k)}\}^{\nu_{ki}}= 0,\tag{1}$ where $$B_ i(z)$$ are polynomials of degree $$\deg B_ i(z)= b_ i$$. The paper contains a list of conditions under which the equation (1) has (can have) rational solutions (2) $$w(z)= P(z)/Q(z)$$, where $$P(z)= a_ p z^ p+\cdots+ a_ 0$$, $$Q(z)= d_ q z^ q+\cdots+ d_ 0$$. The characteristics of rational solutions (2) as the degree $$(p,q)$$, the difference $$p-q$$ and the ratio $$a_ p/d_ q$$ are expressed directly in terms of the parameters $$k$$, $$\nu_{ki}$$, $$b_ i$$. The proofs are omitted.
Reviewer: A.Klíč (Praha)
##### MSC:
 34M99 Ordinary differential equations in the complex domain