On a special class of non complete webs. (English) Zbl 1189.14066

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.


14R99 Affine geometry
14C21 Pencils, nets, webs in algebraic geometry
13N99 Differential algebra
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