高等代数学 Chapter 5
Chapter 5.10 Polynomials
Theorem 5.7.4
Eisenstein Method
\[ f(x) = \sum a_i x^i\]
cannot be divided other polynomials if
\[ p \not | a_n, p | a_i,p^2 \not | a_0\] #### Definition 5.10.1 Sylvester Matrix
\[ R(f,g)=\det(\left[\phantom{\begin{matrix}f_0\\ \ddots\\f_0\\b_0\\ \ddots\\b_0 \end{matrix}} \right.\hspace{-1.5em} \underbrace{\begin{matrix} a_m & \cdots & a_0 & \\ \ddots & & \ddots & \\ & a_m & \cdots & a_0 \\ b_n & \cdots & b_0 & \\ \ddots & & \ddots & \\ & b_n & \cdots & b_0 \end{matrix}}_{m+n-i} \hspace{-1.5em} \left.\phantom{\begin{matrix}f_0\\ \ddots\\f_0\\b_0\\ \ddots\\b_0 \end{matrix}}\right]) \]
where
\[ f(x) = \sum_{i=0}^m a_ix^i\]
\[ g(x) = \sum_{i=0}^n b_ix^i\]
Theorem 5.10.1
Check for common roots for two polynomial
\[ R(f,g) \not = 0\]
Theorem 5.10.2
\[ R(f,g) = a_0^nb_0^m \prod_{i=1}^m \prod_{j=1}^n(x_i-y_j)\]
Theorem 5.10.3
\[ \Delta(f) = (-1)^{\frac{n(n-1)}{2}}\frac{R(f,f')}{a_0} = a_0^{2n-2} \prod_{i\leq j}(x_i-x_j)^2\]
If \(\Delta(f) = 0\), then there is no repeated roots.
Recall that \(f(x)\) have no repeated roots iff \(\deg(\gcd(f,f')) = 0\). This can be used to prove 5.10.3.
Exercise 1
\[f(x) = x^3+3x^2-x+4\]
\[g(x) = x^2-2x-1\]
\[R(f,f') = \det(\begin{bmatrix} 1 & 3 & -1 & 4 & 0 \\ 0 & 1 & 3 & -1 & 4 \\ 1 & -2 & -1 & 0 & 0 \\ 0 & 1 & -2 & -1 & 0 \\ 0 & 0 & 1 & -2 & -1 \end{bmatrix}) = \boxed{161}\]
Exercise 2
Find delta. \[ f(x) = x^3 + px + q\]
\[ f'(x) = 3x + p\]
\[R(f,f') = \det(\begin{bmatrix} 1 & 0 & p & q & 0 \\ 0 & 1 & 0 & p & q \\ 3 & 0 & p & 0 & 0 \\ 0 & 3 & 0 & p & 0 \\ 0 & 0 & 3 & 0 & p \end{bmatrix}) = 27q^2+4p^3\]
\[ \boxed{\Delta(f) = (-1)^3(27q^2+4p^3)}\]
Exercise 3.1
Proof
\[R(f,g) = (-1)^{mn}R(g,f)\]
\[R(f,g) = a_0^nb_0^m \prod_{i=1}^m \prod_{j=1}^n(x_i-y_j) = a_0^nb_0^m (-1)^{mn} \prod_{i=1}^m \prod_{j=1}^n(y_j-x_i) = R(g,f)\]
Exercise 3.2
Proof
\[ R(af,bg) = a^m b^nR(f,g) \]
\[ R(af,bg) = a^na_0^nb^mb_0^m \prod_{i=1}^m \prod_{j=1}^n(x_i-y_j) = a^m b^nR(f,g)\]
Exercise 5
Proof
\[ R(f,g_1g_2) = R(f,g_1)R(f,g_2) \]
\[ R(f,g_1g_2) = a_0^{m+n} \prod_{i=1}^{t} g(x_i) \]
\[ R(f,g_1) = a_0^{m} \prod_{i=1}^t g_1(x_i) \]
\[ R(f,g_2) = a_0^{n} \prod_{i=1}^t g_2(x_i) \]
Exercise 6
Let \(a_0 = 1\). Given \(\Delta(f(x))\). Find \(\Delta(f(x^2))\).
\[\Delta(f(x)) = \prod_{i\leq j}(x_i-x_j)^2\]
\[\Delta(f(x^2)) = \prod_{i\leq j}(\sqrt{x_i}-\sqrt{x_j})^2(\sqrt{x_i}+\sqrt{x_j})^2(-\sqrt{x_i}+\sqrt{x_j})^2(-\sqrt{x_i}-\sqrt{x_j})^2\]
\[\Delta(f(x^2)) = \prod_{i\leq j}(\sqrt{x_i}-\sqrt{x_j})^4(\sqrt{x_i}+\sqrt{x_j})^4\]
\[\Delta(f(x^2)) = \prod_{i\leq j}(x_i-x_j)^4 = \boxed{\Delta(f(x))^2}\]
Exercise 7
Find an equation for curve:
\[ x = t^3 + 2t -3 \]
\[ y = t^2-t +1 \]
\[ f(t) = t^3 + 2t -3 -x \]
\[ g(t) = t^2-t +1 - y \]
so for a given \((x,y)\), there should exist a \(t\) such that both \(f(t)\) and \(g(t)\) both is \(0\). So they have a common root.
\[ R(f,g) = \det(\begin{bmatrix} 1 & 0 & 2 & -3-x & 0 \\ 0 & 1 & 0 & 2 & -3-x \\ 2 & -1 & 1-y & 0 & 0 \\ 0 & 2 & -1 & 1-y & 0 \\ 0 & 0 & 2 & -1 & 1-y \end{bmatrix}) \]
\[ R(f,g) = \boxed{y^3 - x^2 + 3xy + y^2 -6x + 10y -12}\]