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 limacon 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.
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