Sylvester矩阵、子结式、辗转相除法的三者关系的应用—

Sylvester矩阵、子结式、辗转相除法的三者关系的应用——Habicht定理

【Habicht定理】

设 $\mathbb{R}$ 为域，非 $0$ 多项式 $a_{0}x^{n + 1} + a_{1}x^{n - 1} + \ldots + a_{n + 1}、G = b_{0}x^{n} + b_{1}x^{n - 1} + \ldots + b_{n}\mathbb{\in R}\lbrack x\rbrack$ 。设 $S_{n + 1} = A、S_{n} = B、S_{n - 1}\ldots S_{1}、S_{0}$ ，为 $A 、 B$ 的子结式链，而 $R_{n + 1}\ldots R_{0}$ 为其主子结式系数链。那么对每个 $1\ldots n$ 都有

$R_{j + 1}^{2(j - i)}S_{i} = subres_{i}\left( S_{j + 1},S_{j} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \leq i < j$

【证明思路】

当 $n = 7 、 j = 5 、 i = 3$ 时，根据以前的结论

$S_{6} = detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{0} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix} \end{pmatrix} = detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{pmatrix} \end{pmatrix}$

$R_{6} = b_{7}b_{7}c_{6}$

$S_{5} = detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{1} \\ F_{0} \\ F_{0} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix} \end{pmatrix} = detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{pmatrix} \end{pmatrix}$

$S_{3} = detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{0} \\ F_{0} \\ F_{0} \\ F_{0} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 0 & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 0 \\ 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix} \end{pmatrix} = detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{2} \\ F_{2} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{pmatrix} \end{pmatrix}$

观察可知

$R_{j + 1}^{2(j - i)}S_{i} = R_{6}^{2 \times 2}S_{3} = \left( b_{7}b_{7}c_{6} \right)^{4}detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{2} \\ F_{2} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{pmatrix} \end{pmatrix} = \left( b_{7}b_{7}c_{6} \right)^{4} \times b_{7}^{2}c_{6}^{2}detpol\begin{pmatrix} d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{pmatrix} = b_{7}^{10}c_{6}^{6}detpol\begin{pmatrix} d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{pmatrix}$

$subres_{i}\left( S_{j + 1},S_{j} \right) = subres_{3}\left( S_{6},S_{5} \right) = \left( b_{7}b_{7} \right)^{5 - 3}\left( b_{7}b_{7}c_{6}c_{6} \right)^{6 - 3}detpol\begin{pmatrix} c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{pmatrix} = b_{7}^{10}c_{6}^{6}detpol\begin{pmatrix} d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{pmatrix}$