Singular solutions of polynomial webs in the plane.
The authors apply differential algebra to the study of singular solutions of first order differential equations with complex polynomial coefficients. Results obtained were either not previously known, or only known by analytic methods. These results are then applied to the theory of (polynomial) planar webs. The algebraic context is the ring \(\mathcal F \{y\}\) of differential polynomials in the differential indeterminate \(y\) over the field of complex rational function \(\mathcal F\). If \(P\) is a differential polynomial of order \(m\), its separant \(S_P\) is the partial of \(P\) with respect to \(y^{(m)}\). When \(P\) is of order one and reduced (no multiple factors), let \(R\) denote the \(y^\prime\) resultant of \(P\) and \(S_P\). The singular solutions of \(P=0\) are the irreducible components of \(R\). The authors prove that a prime differential ideal is a component of singular solutions of \(P\) if and only if the ideal contains both \(P\) and \(S_P\). Moreover, any such prime ideal is generated, as a radical differential ideal, by an irreducible component of \(R\). For polynomial Clairaut equations, the authors give necessary and sufficient conditions for the existence of a unique component of singular solutions. Finally, the authors show that there is a correspondence between irreducible polynomial webs and prime differential ideals, and conclude therefore that the components of singular solutions are invariants of polynomial webs. Applications of this are presented.
scientific article; zbMATH DE number 5144396