http://vmath.ru/vf5/ 2026-10-13T03:43:21+0300 http://vmath.ru/vf5/ http://vmath.ru/vf5/lib/tpl/vf5/images/favicon.ico text/html 2023-07-12T19:51:33+0300 Anonymous (anonymous@undisclosed.example.com) http://vmath.ru/vf5/detse/discrime?rev=1689180693&do=diff Author---Русская версия ---------- Discriminant Denote by $ \mathbb A_{} $ any of the sets $ \mathbb Z_{}, \mathbb Q, \mathbb R $ or $ \mathbb C_{} $. Discriminant of the polynomial $$ f(x)=a_{0}x^n+a_1x^{n-1}+\dots+a_n, \ (n>1, a_0\ne 0) $$ coincides with the $$ {\mathcal D}(f)=\frac{(-1)^{n(n-1)/2}}{a_0}{\mathcal R}(f(x),f^{\prime}(x)) \ . $$$ \lambda_{1},\dots,\lambda_n $$ f_{}(x) $$$ {\mathcal D}(f) =(-1)^{n(n-1)/2} a_0^{n-2}\prod_{j=1}^n f^{\prime}(\lambda_j)= a_0^{2n-2} \prod_{1\le … text/html 2021-11-24T00:16:11+0300 Anonymous (anonymous@undisclosed.example.com) http://vmath.ru/vf5/detse/resultante?rev=1637702171&do=diff Author Русская версия --- ---------- Resultant Denote by $ \mathbb A_{} $ any of the sets $ \mathbb Z, \mathbb Q, \mathbb R_{} $ or $ \mathbb C_{} $. Resultant in the Sylvester form For the polynomials $ f(x)=a_{0}x^n+a_1x^{n-1}+\dots+a_n $ and $ g(x)=b_{0}x^m+b_1x^{m-1}+\dots+b_m $ при $ a_{0}\ne 0, b_0\ne 0 $ consider the square matrix of the order $ m+n_{} $ $$ M=\left(\begin{array}{cccccccccc} a_0&a_1&a_2&\ldots&\ldots&a_n&0&\dots &0 &0\\ 0&a_0&a_1&\ldots&\ldots&a_{n-1}&a_n&\dots&0 …