## 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.

### MSC:

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

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