On the polynomial representation of generalized Liouville operators (Q1186716)

The generalized Liouville operators \({\mathcal L}_ m F=\text{det}\| F_{ij}\|^ m_{i,j=0}\), where \(F_{ij}=\partial^{i+j}F(x,y)/\partial x^ i \partial y^ j\) (\(F(x,y)\) being any infinitely differentiable two-variable function) is considered. The problem of finding a polynomial representation for \({\mathcal L_ n}({\mathcal L}_ m F)\) by the means of algebraic manipulations is analysed. Using computer programs the authors expand the Liouville expressions into sums of individual terms via direct computation. The algorithms recognize groups of isolated terms which cannot be represented completely as polynomials over the Liouville operators. It is shown that no general polynomial expression can exist for the composite operation \({\mathcal L}_ n({\mathcal L}_ m F)\).