1. Home
  2. The art of the pfug
  3. Algebraic geometry
  4. A brief introduction to

1.1 A brief introduction to computational algebraic geometry  (Page 2/20)

<< Chapter < Page
  The art of the pfug     Page 2 / 20
Chapter >> Page >
y + 1 2 x - 3 + y + 1 x - 3 2 + 3 y + 1 2 + 4 y + 1 x - 3 - x - 3 2 - 5 x + 3 y + 12

where here we are asking Sage to write x 2 y + x y 2 as a series in x - 3 and y + 1 . The 10 in the command is telling Sage to only compute the terms of the series up to degree 10, though we know that really we're dealing with a polynomial of degree 3, so higher degree terms in x - 3 and y + 1 couldn't possibly show up in the Taylor expansion (Why?).

Exercises

For these exercises, I would recommend using the Sage computer algebra system ( http://www.sagemath.org/ ), a free open-source alternative to Maple, Mathematica, etc.; for example, to graph the solution set to x 2 + y 2 = 2 in Sage, we might use the commands:

var("x y") implicit_plot(x^2+y^2-2, (x,-3,3), (y,-3,3)).show(aspect_ratio=1)

To get inline help on a command in Sage, use that command followed by a ?, as in:

implicit_plot? taylor?
  1. For each of the following curves, calculate the partial derivatives and find the singular points. Then calculate the Taylor expansion for the curve about eachsingular point. If possible, identify each singularity as a cusp or a node. Finally plot each curve.
    1. x 2 = x 4 + y 4
    2. x y = x 6 + y 6
    3. x + y = x 3 - y 3
    4. x 3 = y 2 + x 4 + y 4
    5. x 2 y + x y 2 = x 4 + y 4
    6. y 2 = x 3 + x 2
    7. ( x 2 + y 2 ) 2 = x 2 - y 2
    8. x 3 + x y 2 + 1 = x + x 2 + y 2
    9. y 2 = x 5 - 2 x 4 + x 3
  2. Prove that y 2 = x 3 + p x + q has no singular points if and only if f ( x ) = x 3 + p x + q has three distinct roots. [Hint: First show that a polynomial f ( x ) has a multiple root at a if and only if f ( a ) = f ' ( a ) = 0 .]

Multiplicity and the tangent cone

We've now seen plane curves with various singularities and given a few types of them names, although we haven't give any precise mathematical definitions of these yet.

Some singularities we've seen

One simple question we should expect to be able to answer using calculus is what are the tangent lines to the branch or branches of the curve at the singular point? (In the node example above, we can see two distinct tangent lines in the picture, whereas in the cusp and tacnode cases, there should only be one tangent line at the singular point.)

To simplify our computations, we may as well first make a change of coordinates so that the singular point is at the origin (we can certainly do this with an affine change of variables–see the homework). Consider then a plane curve V ( f ) , where f is a polynomial with f ( 0 , 0 ) = 0 . The notation V ( f ) will be used to mean { ( x , y ) ∈ R 2 : f ( x , y ) = 0 } . More generally, an algebraic variety over the real numbers R will be the locus

V ( f 1 , f 2 , ... , f k ) = ( x 1 , ... , x n ) ∈ R n : f 1 ( x 1 , ... , x n ) = f 2 ( x 1 , ... , x n ) = ⋯ = f k ( x 1 , ... , x n ) = 0
of common zeros of finitely many polynomials in n variables f 1 , ... , f k ∈ R [ x 1 , ... , x n ] .

Of course, if the origin were a smooth point, this would be easy: there is only one tangent line, defined by

∂ f ∂ x x + ∂ f ∂ y y = 0 .

When the origin is singular, both first partial derivatives are zero and we need to look at higher order terms to find the tangent lines. In particular, let us write f = f m + f m + 1 + ... + f n where f k is the degree k part of the polynomial f ; here n = deg f and f m ≠ 0 is the non-zero piece of f of smallest degree.

Proposition If L = V ( a x + b y ) is a tangent line to V ( f ) at p = ( 0 , 0 ) , then a x + b y is a factor of f m .

Let ( x ( t ) , y ( t ) ) for t ∈ [ 0 , 1 ) be a local parametrization of a branch of X = V ( f ) near the origin, with x ( 0 ) = y ( 0 ) = 0 . We assume for convenience that the branch does not have a vertical tangent, i.e. that y ( t ) / x ( t ) is bounded as t → 0 . Then the slope of the tangent line to this branch of X at the origin should be lim t → 0 + y ( t ) x ( t ) . We write

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

Ask