Parametric expansions of an algebraic variety near its singularities

This talk is devoted to the application of methods of power geometry to find asymptotic and asymptotic expansions of solutions of nonlinear algebraic equations near its singularities. Power geometry algorithms, its implementation and applications to feature resolution based on superpositions of degree transformations given by unimodular matrices will be described. A vector generalization of chain fractions based on degree transformations and Euler’s algorithm will be described. In the study of invariant Einstein’s metrics associated with Ricci flows, there arises an algebraic manifold $\Omega$ which is described by a 12th order algebraic equation of three variables. This manifold has several isolated special points as well as special curves. Local parametric expansions of the algebraic manifold near all special points and curves of special points are calculated by methods of power geometry. With their help near special points and curves of special points the structure of the manifold $\Omega$ is studied both in a finite part of space and in infinity. For each case, the truncated equation is computed and Newton’s polyhedron is constructed. If the truncated equations contain nonlinear multipliers, the appropriate degree transformation of the variables is applied.