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Optimization of Wu's algorithm for the elimination of polynomial variables by High-Performance Computing (HPC)

Abstract:

Elimination theory, which is the algorithmic representation to eliminate indeterminates between polynomials of several indeterminates, is considered the origin of algebraicgeometric relation [1]. The theory deals with the task of finding the complete solutions of a system of algebraic equations f1 (x1,..., xn)= 0, f2 (x1,..., xn)= 0,..., fs (x1,..., xn)= 0,(1.1) where f1,..., fs∈ K [x]. The solutions derived from each polynomial set depend on the number of indeterminates; thereby, in one indeterminate, the solutions are entirely known. Hence, the fundamental process behind the elimination theory is to reduce the system of equations into an equivalent system of equations in fewer indeterminates. Consequently, algorithms that solve polynomial systems efficiently have been widely studied over time. Euler (1748) and Bézout (1764) solved systems of two polynomials in one indeterminate. Bézout’s work coined the term Resultants 1, and it was used in computations based on methods developed in the early century. For example, works of Sylvester, Macaulay, Cayley, and Dixon were included in the category of Resultants methods. These works were considered the only and essential tools to compute The zeros of a function is any element that replacement the indeterminate that will produce an answer of zero, see Annex A 7.(Zeros) of a set of polynomials. However, algebraic equations systems become more complex to solve, for example, increasing the number of