×

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
PDF BibTeX XML Cite
Full Text: DOI Numdam Numdam EuDML

References:

[1] Beauville (A.).— Géométrie des tissus [d’après S. S. Chern et P. A. Griffiths], Séminaire Bourbaki (1978/79), Exp. No. 531, p. 103-119, Lecture Notes in Math., 770, Springer, Berlin, (1980). · Zbl 0436.57008
[2] Freudenburg (G.).— Algebraic theory of locally nilpotent derivations, Encyclopaedia of Mathematical Sciences, 136, Invariant Theory and Algebraic Transformation Groups, VII, Springer-Verlag, Berlin, (2006). · Zbl 1121.13002
[3] Hénaut (A.).— Sur la linéarisation des tissus de \(\mathbf{C}^2\), Topology 32, no. 3, p. 531-542 (1993). · Zbl 0799.32010
[4] Hénaut (A.).— On planar web geometry through abelian relations and connections, Ann. of Math. (2) 159, no. 1, p. 425-445 (2004). · Zbl 1069.53020
[5] Miyanishi (M.).— Vector fields on factorial schemes, J. Algebra 173, no. 1, p. 144-165 (1995). · Zbl 0835.13006
[6] Ripoll (O.).— Géométrie des tissus du plan et équations différentielles, Thèse de doctorat, Université Bordeaux 1, décembre 2005, available on http://tel.archives-ouvertes.fr/tel-00011928.
[7] Ripoll (O.), Sebag (J.).— Solutions singulières des tissus polynomiaux du plan, J. Algebra 310, no. 1, p. 351-370 (2007). · Zbl 1141.53013
[8] Ripoll (O.), Sebag (J.).— The Cartan-Tresse linearization polynomial and applications, Journal of Algebra, Volume 320, no. 5, p. 1914-1932 (2008). · Zbl 1154.13009
[9] Ripoll (O.), Sebag (J.).— Tissus du plan et polynômes de Darboux, Ann. Fac. Sci. Toulouse, 19, no. 1, p. 1-11 (2010).
[10] Ripoll (O.), Sebag (J.).— Nilpotent webs, to appear in Journal of Commutative Algebra.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.