Versal deformations of differential forms of degree \(\alpha\) on a line
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.
scientific article; zbMATH DE number 3915260