On a special class of non complete webs (Q2380710)

A \(d\)-web is given by a differential equation \(F(x,y,y')=0\), where \(F(x,y,p)\in {\mathbb C}\{x,y\}[p]\). An algebraic \(d\)-web is determined by a polynomial \(G\in {\mathbb C}[s,t]\) via the Legendre transformation \(F(x,y,p)=G(y-px,p)\). The derivation \(\partial _x+p\partial _y\) is locally nilpotent, i.e. for every \(f\in {\mathbb C}[x,y,p]\) there exists \(n\in {\mathbb N}\) such that \(d_F^n(f)=0\) and \((\partial _x+p\partial _y)(F)=0\) in the algebraic case. The author considers non complete webs defined by polynomials only in \(y\) and \(p\). He answers the question what nilpotence means in that context.