Finite energy monopoles in non-Abelian gauge theories on odd-dimensional spaces

In higher-dimensional gauge theory, we need energies with higher power terms of field strength in order to realize pointwise monopoles. We consider new models with higher power terms of field strength and extraordinary kinetic terms of the scalar field. Monopole charges are computed as integrals over spheres and they are related to mapping class degree. Hedgehog solutions are investigated in these models. Every differential equation for these solutions is Abel's differential equation. A condition for the existence of a finite energy solution is shown. The spaces of 1-jets of these equations are defined as sets of zeros of polynomials. Those spaces can be interpreted as singular quartic surfaces in three-dimensional complex projective space.