Solution of elementary equations in the Minkowski geometric algebra of complex sets

Abstract

The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks $\mathcal{A}$ and ℬ with radii a and b, a solution of the linear equation $\mathcal{A}\otimes \mathcal{X}=\mathcal{B}$ in an unknown set $\mathcal{X}$ exists if and only if a≤b. When it exists, the solution $\mathcal{X}$ is generically the region bounded by the inner loop of a Cartesian oval (which may specialize to a limaçon of Pascal, an ellipse, a line segment, or a single point in certain degenerate cases). Furthermore, when a<b<1, the solution of the nonlinear monomial equation $\mathcal{A}\otimes(\otimes^{n}\mathcal{X})=\mathcal{B}$ is shown to be the region that is bounded by a single loop of a generalized form of the ovals of Cassini. The latter result is obtained by considering the nth Minkowski root of the region bounded by the inner loop of a Cartesian oval. Preliminary consideration is also given to the problems of solving univariate polynomial equations and multivariate linear equations with complex disk coefficients.