Relations Between Roots and Coefficients, Interpolation and Application to System Solving

Abstract

We propose an algebraic framework to represent zero-dimensional algebraic systems. In this framework, we give new interpolation formulae. We use this good representation of the algebraic systems to develop a generalization of Weierstrass's method to the multivariate systems. This method allows us to approximate simultaneously all the roots of an algebraic system. We obtain an effective iteration function with a quadratic convergence in a neighbourhood of the solutions. We use this Weierstrass iteration function in a continuation method to obtain a global method. Experiments are exposed to underline the efficiency of the approach.

Efficient Computation of Algebraic Immunity for Algebraic and Fast Algebraic Attacks

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Solving Systems of Polynomial Equations: a Novel End Condition and Root Computation Method

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“…The most popular iterative method is Newton's method and it works well locally and only if initial point is a good guess, which is difficult in solving systems of polynomial equations. Other methods are Newton like methods, minimization methods or Weierstrass method [12], [13].…”

Efficient erasure list-decoding of Reed-Muller codes

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