论文标题
分解双度的四个离子多项式（n，1）
Factorization of Quaternionic Polynomials of Bi-Degree (n,1)
论文作者
论文摘要
我们考虑了双度$（n，1）$的多项式在四季度的偏斜场上，在这些四偏差范围内，不确定的彼此和所有系数都相互通勤。这种类型的多项式通常不承认因素化。我们回想起存在单变线性因子的分解的必要条件，这最初由Skopenkov和Krasauskas说明。通常，这种分解是通过已知分解季节性多项式的已知分解结果的非唯一分解。我们揭示了具有非唯一因素化的双变量多项式存在，这些因素无法以这种方式解释，并以几何和代数来表征它们。因素化的存在与在季节上的投影空间中的两种二元多项式参数的统治表面上存在两种不同类型（左/右）的特殊裁决有关。从上述意义上讲，特殊的非唯一性可以通过适当因素化因素的换向特性来解释代数。发生这种情况的必要几何条件是退化至至少一个左/右裁决之一。
We consider polynomials of bi-degree $(n,1)$ over the skew field of quaternions where the indeterminates commute with each other and with all coefficients. Polynomials of this type do not generally admit factorizations. We recall a necessary and sufficient condition for existence of a factorization with univariate linear factors that has originally been stated by Skopenkov and Krasauskas. Such a factorization is, in general, non-unique by known factorization results for univariate quaternionic polynomials. We unveil existence of bivariate polynomials with non-unique factorizations that cannot be explained in this way and characterize them geometrically and algebraically. Existence of factorizations is related to the existence of special rulings of two different types (left/right) on the ruled surface parameterized by the bivariate polynomial in the projective space over the quaternions. Special non-uniqueness in above sense can be explained algebraically by commutation properties of factors in suitable factorizations. A necessary geometric condition for this to happen is degeneration to a point of at least one of the left/right rulings.
