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Versal deformations of differential forms of degree \(\alpha\) on a line. (English. Russian original) Zbl 0573.58002

Funct. Anal. Appl. 18, 335-337 (1984); translation from Funkts. Anal. Prilozh. 18, No. 4, 81-82 (1984).
A family \(F(x,\lambda)(dx)^{\alpha}\) is called a versal deformation of the form \(f(x)(dx)^{\alpha}\) if F(x,0)\(\equiv f(x)\) and every family \(G(x,\epsilon)(dx)^{\alpha}\) satisfying G(x,0)\(\equiv f(x)\) can be expressed as \[ G(x,\epsilon)\equiv F(H(x,\epsilon),\phi (\epsilon))\quad (\partial H(x,\epsilon)/\partial x)^{\alpha} \] with appropriate germs H,\(\phi\) (H(x,0)\(\equiv x\), \(\phi (0)=0)\). Result: Let \(\mu\) be a nonnegative integer. If \(\alpha\) \(\neq -(\mu +1)/q\) for any natural q, then the family \((x^{\mu +1}+\lambda_ 1x^{\mu - 1}+...+\lambda_{\mu -1}x+\lambda_{\mu})(dx)^{\alpha}\) is a versal deformation of the form \(x^{\mu +1}(dx)^{\alpha}\). If \(\alpha =-(\mu +1)/q\) (q natural), then \((x^{\mu +1}+\lambda_ 1x^{\mu - 1}+...+\lambda_{\mu}+\lambda x^{\mu +q})(dx)^{\alpha}\) is a versal deformation of the same form.
Reviewer: J.Chrastina

MSC:

58A10 Differential forms in global analysis
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[1] V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings. Classification of Critical Points, Caustics and Wave Fronts [in Russian], Nauka, Moscow (1982).
[2] V. I. Arnol’d, Itogi Nauki i Tekhniki, Sov. Probl. Mat.,22, 3-55 (1983).
[3] A. B. Givental?, Usp. Mat. Nauk,38, No. 6, 109-110 (1983).
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