On the Zeroes of Clifford Algebra-Valued Polynomials with Paravector Coefficients

Abstract

In this paper, we first show the zeros of monic Clifford algebra polynomials with paravector coefficients by generalizing a right-division algorithm for quaternion polynomials. Then we provide a necessary and sufficient condition for the polynomials, which have an infinite number of roots. In particular, we derive the explicit formulas for computing the zeros of monic Clifford algebra quadratic polynomials with paravector coefficients and some cubic, quartic polynomials with real coefficients. From this, we obtain that any monic Clifford algebra quadratic polynomial with paravector coefficients has at least one zero.