5
$\begingroup$

I am totally stack in how to do this exercise : How can I find the irreducible components of $ V(X^2 - XY - X^2 Y + X^3) $ in $A^2$(R)?

Given an algebraic set, what is the consideration I have to make to find them? I know the definitions of components and irreducible components but these don't help in to solve this kind of exercise in the end.

Can anyone suggest a good reference book also?

Thanks

$\endgroup$
5
$\begingroup$

Write your polynomial as a product of irreducible polynomials. The vanishing loci of these irreducible polynomials are then the irreducible components of your variety.

| cite | improve this answer | |
$\endgroup$
2
$\begingroup$

$V(f)$, the zero set of a polynomial $f \in R[x,y]$, is given as set of points such that $f(x,y) = 0$. If this polynomial is reducible, i.e. $f=gh$, the points $(x_0, y_0)$ in $V(f)$ are such that $g(x_0, y_0) = 0$ or $h(x_0, y_0) = 0$. This means that $V(f)$ is a union of $V(g)$ and $V(h)$.

In your case, $f(x,y)=x^2-xy-x^2y+x^3=x(x+1)(x-y)$. Zeros of this polynomial are all the points for which any factor of $f$ is zero. Thus we see that $V(f) = V(x) \cup V(x+1) \cup V(x-y)$. Geometrically, we can imagine this in $\mathbb{R}^2$ as a union of three lines, the y-axis, diagonal line through the origin passing through the first quadrant and parallel to the y axis passing through the point $(-1,0)$.

As for the books, very nice introductory book about algebraic curves which is freely available is by Fulton: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf. Quick google search also turned up these notes which seem to contain some nice examples and theorems ilustrating what is going on: http://csclub.uwaterloo.ca/~mlbaker/get.php?name=LW-1135-pmath764notes.pdf.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Would this change if we consider $A^2(C)$? I don't think we have used anywhere the fact that we are in the field of real numbers $\endgroup$ – User Mar 23 '14 at 9:34
  • $\begingroup$ You are right, this will work in C as well. Actually I think this will work in any field (not just characteristic 0). $\endgroup$ – BoZenKhaa Mar 23 '14 at 9:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.