Geodesic
Maslov Dequantization and the Homotopy Method for Solving Systems of Nonlinear Algebraic Equations
Matematičeskie zametki, Tome 83 (2008) no. 2, pp. 221-231

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The Maslov dequantization allows one to interpret the classical Gräffe–Lobachevski method for calculating the roots of polynomials in dimension one as a homotopy procedure for solving a certain system of tropical equations. As an extension of this analogy to systems of $n$ algebraic equations in dimension $n$, we introduce a tropical system of equations whose solution defines the structure and initial iterations of the homotopy method for calculating all complex roots of a given algebraic system. This method combines the completeness and the rigor of the algebraic-geometrical analysis of roots with the simplicity and the convenience of its implementation, which is typical of local numerical algorithms.
Mots-clés : Maslov's dequantization
Keywords: Gräffe–Lobachevski method, tropical equations, complex roots, tropical surface, amoeba of a surface, spine of an amoeba. 
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     author = {B. Kh. Kirshtein},
     title = {Maslov {Dequantization} and the {Homotopy} {Method} for {Solving} {Systems} of {Nonlinear} {Algebraic} {Equations}},
     journal = {Matemati\v{c}eskie zametki},
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B. Kh. Kirshtein. Maslov Dequantization and the Homotopy Method for Solving Systems of Nonlinear Algebraic Equations. Matematičeskie zametki, Tome 83 (2008) no. 2, pp. 221-231. http://geodesic.mathdoc.fr/item/MZM_2008_83_2_a5/