Noncommutative Vieta Theorem in Clifford Geometric Algebras

TL;DR

This paper extends Vieta's formulas to Clifford geometric algebras, providing a basis-free determinant formula and generalizing the theorem for arbitrary dimensions, with applications in computational fields.

Contribution

It introduces a basis-free determinant formula and generalizes Vieta's theorem to all dimensions in geometric algebra, connecting noncommutative algebra with classical polynomial relations.

Findings

Generalized Vieta's formulas for Clifford algebras

Existence of a basis-free determinant formula in geometric algebra

Application potential in computational sciences

Abstract

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta's formulas with the ordinary Vieta's formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand--Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis-free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering.