Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, (2) separating truncated equations, and (3) power transformations of coordinates. Here, we describe the method for the simplest case, a single algebraic equation, and apply it to an algebraic variety, as described by an algebraic equation of order 12 in three variables. The variety was considered in study of Einstein’s metrics and has several singular points and singular curves. Near some of them, we compute a local parametric expansion of the variety.
Here we give an algorithm for solving the following problem. Let m<n integer vectors be given in the n-dimensional real space. Their linear span forms a linear subspace L in R<sup>n</sup>. It is required to calculate such an unimodular matrix that a linear transformation with it transforms the subspace L into a coordinate one. Also, programs that implement the algorithms and power transformations, for which they are needed, are given.
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