Solving Systems of Polynomial Equations - A Tensor Approach.

Polynomial relations are at the heart of mathematics. The fundamental problem of solving polynomial equations shows up in a wide variety of (applied) mathematics, science and engineering problems. Although different approaches have been considered in the literature, the problem remains difficult and requires further study. We propose a solution based on tensor techniques. In particular, we build a partially symmetric tensor from the coefficients of the polynomials and compute its canonical polyadic decomposition. Due to the partial symmetry, a structured canonical polyadic decomposition is needed. The factors of the decomposition can then be used for building systems of linear equations, from which we find the solutions of the original system. This paper introduces our approach and illustrates it with a detailed example. Although it cannot solve any system of polynomial equations, it is applicable to a large class of sub-problems. Future work includes comparisons with existing methods and extending the class of problems, for which the method can be applied.