Algebraic differential equations with a maximal number of polynomial solutions of given structure. (Russian) Zbl 0668.34008

Lukashevich, N. A.; Deniskovets, A. A.; Nemets, V. S.

Differ. Uravn. 24, No. 12, 2172-2174 (1988).

Let G(z) and H(z) be polynomials of complex variable, z and \(\epsilon_ t\) roots of the equation \(\epsilon^{\delta}=1\), \(\delta\in N\). P. R. Lazov [Approximate methods for investigating, differential equations and their applications, Kujbyshev 1979, 112-118 (1979; Zbl 0485.34005)] studied when all polynomials of the family \(\omega =\epsilon_ tG(z)+H(z),\) \(t=1,...,\delta\) are solutions of algebraic differential equations of special forms. Similar questions are here considered for algebraic equations \(\sum^{N}_{i=0}B_ i(z)\prod^{S_ i}_{k=1}\{\omega^{(\ell_{ki})}\}^{V_{k_ i}}=0\) and \(\sum^{T}_{i=0}A_ i(z)\omega^{(\gamma_ i)}+\sum^{T}_{\ell =0}B_{\ell}(z)\{\omega^{(\rho)}\}^{\lambda_{\ell}}=0.\)

Reviewer: J.H.Tian

MSC:

34M99	Ordinary differential equations in the complex domain
34C99	Qualitative theory for ordinary differential equations

Keywords:

algebraic differential equations; maximal number of polynomial solutions; algebraic equations