AI 中文总结
该研究提出二元四元数多项式的分解条件，将其应用于天体曲面分解，改进了相关定理并扩展至四次情形，证明特定天体曲面无法同时为圆的和与积。
AI 中文摘要
我们针对任意双次数的二元四元数多项式，给出其存在单变量线性左因子或右因子的代数与几何条件。将该四元数分解定理应用于双次数(1,1)情形，可得到椭圆几何中Clifford的经典定理；应用于双次数(2,2)情形，我们得到天体曲面的双圆分解，这类曲面是三维球面中经过一般点包含两个圆的曲面。这一结果为Skopenkov与Krasauskas 2019年的定理提供了另一种证明并加以改进，该定理指出非四次天体曲面莫比乌斯等价于单位四元数中圆的逐点乘积，或欧氏空间中圆的逐点和的逆球极投影。我们提出的方法将该分解结果扩展到四次情形，且证明了在莫比乌斯等价与球极投影下，曲面不能同时是圆的和与积。
英文摘要
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.
