Abstract:
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.
Keywords:Maslov's dequantization, Gräffe–Lobachevski method, tropical equations, complex roots, tropical surface, amoeba of a surface, spine of an amoeba.
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