Algorithms for Solving an Algebraic Equation

Bruno, A D

doi:10.1134/s0361768819100013

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2023

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Cited by 5 publications

(2 citation statements)

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“…The equation h k = 0 has neither a polynomial solution nor a parametric one. Then, using Hadamard's polyhedron [15], one can compute a piecewise approximate parametric solution to the equation h k = 0 and look for an approximate parametric expansion.…”

Section: Casementioning

confidence: 99%

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Parametric Expansions of an Algebraic Variety near Its Singularities

Bruno

Azimov

2023

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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.

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Section: Casementioning

confidence: 99%

“…3 and curves F , I and K. In Sections 7-10, the neighborhoods of points P 3 are studied, correspondingly. The methods proposed in [15] and described in Sections 2-5 are implemented.…”

Section: Namementioning

confidence: 99%