Bivariate quaternionic factorizations and surfaces that decompose into two circles
We present an algebraic and geometric condition for bivariate quaternionic polynomials of arbitrary bidegree to have a univariate linear left or right factor. We apply this quaternionic factorization theorem to the bidegree (1,1) case and recover a classical theorem of Clifford in elliptic geometry. By applying to the bidegree (2,2) case, we obtain decompositions into two circles of celestial surfaces, namely surfaces in the 3-dimensional sphere that contain two circles through a general point. This results in an alternative proof and refinement for a theorem by Skopenkov and Krasauskas from 2019, which states that a non-quartic celestial surface is Möbius equivalent to either the pointwise product of circles in the unit-quaternions, or an inverse stereographic projection of the pointwise sum of circles in Euclidean space. Our proposed method extends this decomposition result to the quartic case and we show that surfaces are, up to Möbius equivalence and stereographic projections, not both a sum and product of circles.