Maslov Dequantization and the Homotopy Method for Solving Systems of Nonlinear Algebraic Equations

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.