1.1 A brief introduction to computational algebraic geometry  (Page 19/20)

To prove that the Vandermonde determinant is equal to the product above, we can apply row and column operations to compute the determinant:

where in the last step we are subtracting from each column multiples of earlier columns. The desired formula for the Vandermonde determinant then follows by induction on $n$ .

Exercises

1. If $f,g\in \mathbf{C}\left[x\right]$ are non-constant polynomials over the complex numbers, show that $f$ and $g$ have a common root in $\mathbf{C}$ if and only if $\text{Res}(f,g,x)=0$ . [Hint: Use the “fundamental theorem of algebra”: the fact that any non-constant polynomial in $\mathbf{C}\left[x\right]$ factors completely into linear factors.]
2. If $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0\in k\left[x\right]$ , where $a_n\ne 0$ and $n>0$ , then the discriminant of $f$ is defined to be
   $$\text{disc}\left(f\right)=\frac{{(-1)}^{n(n-1)/2}}{a_n}\text{Res}(f,f^{\text{'}},x)$$
   Prove that $f$ has a multiple factor (that is, $f$ is divisible by $g^2$ for some non-constant $g\in k\left[x\right]$ ) if and only if $\text{disc}\left(f\right)=0$ .
3. 1. Compute the discriminant of the polynomial $x^2+bx+c$ for $b,c\in k$
   2. Compute the discriminant of the polynomial $x^3+px+q$ for $p,q\in k$ .
4. An alternative definition of the discriminant of a polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0\in \mathbf{C}\left[x\right]$ of degree $n$ is
   $$D\left(f\right)=a_n^{2n-2}\prod _{i<j}{({\alpha }_i-{\alpha }_j)}^2$$
   where ${\alpha }_1,...,{\alpha }_n$ are the roots of $f$ , counted with multiplicities, i.e. ${\displaystyle f\left(x\right)=a_n\prod _i(x-{\alpha }_i)}$ .
   1. Show that $D\left(f\right)$ doesn't depend on the choice of ordering on the roots ${\alpha }_i$ .
   2. Show that ${\displaystyle \prod _{i<j}{({\alpha }_i-{\alpha }_j)}^2={(-1)}^{n(n-1)/2}\prod _{i\ne j}({\alpha }_i-{\alpha }_j)}$ .
5. For $f,g\in \mathbf{Z}\left[x\right]$ , show that $\text{Res}(f,g,x)\in \mathbf{Z}$ .
6. Let $f,g\in k\left[x\right]$ have degrees $n$ and $m$ . Show that if the $(m+n)\times (m+n)$ Sylvester matrix has rank $m+n-1$ , then $f$ and $g$ share a common linear factor, but not a common quadratic factor.

Quotient rings and the tjurina number

Starting with the integers $\mathbf{Z}$ , if we fix a positive integer $n$ , we can construct the integers modulo $n$ as follows: we let

be the set of possible remainders upon dividing an integer by $n$ . To add or multiply $x,y\in \mathbf{Z}/⟨n⟩$ , we add or multiply them in $\mathbf{Z}$ and then take the remainder upon division by $n$ , e.g. we would write $xy=qn+r$ , and the product of $x$ and $y$ in $\mathbf{Z}/⟨n⟩$ would be $r$ . Alternatively, one says that $x$ and $y$ are congruent modulo $n$ and write $x\equiv y(modn)$ if $y-x$ is divisible by $n$ . One can check directly from the definition this relation satisfies the following properties (where $\equiv$ denotes congruence modulo a fixed number $n$ ):