1.1 A brief introduction to computational algebraic geometry (Page 21/20)

The art of the pfug Page 1 / 20

A brief introduction to algebraic geometry through singularities of plane algebraic curves and Groebner basis computation.

This is a collection of lecture notes and problem sets from MATH 499: VIGRE Computational Algebraic Geometry at Rice University in 2010.

Plane curve singularities

A point $p$ is a singular point of a plane curve $f (x, y) = 0$ if and only if $f (p) = 0$ so that $p$ is on the curve, and

{(\nabla, f|}_{p} = ({(\frac{\partial f}{\partial x}|}_{p}, {(\frac{\partial f}{\partial y}|}_{p}) = 0 .

In single variable calculus, for studying points where the first derivative of a function $g (x)$ is zero, it is often helpful to study the higher derivatives. For example, if $g^{'} (a) = 0$ and $g^{''} (a) > 0$ , then $g$ has a local minimum at $a$ . More generally, if $g$ is a sufficiently nice function The technical term is “real analytic,” which just means represented by its Taylor series near each point. Essentially all infinitely differentiable functions one encounters in practice are real analytic. Probably the simplest example of a smooth function that isn't real analytic is something like

g (x) = \{\begin{matrix} e^{- 1 / x^{2}}, & if x \neq 0; \\ 0 & if x = 0 . \end{matrix})

You can check that this function is infinitely differentiable, but

g (x) \to 0

so fast as

x \to 0

that

g^{(n)} (0) = 0

for all

n

. Thus its Taylor series centered at

x = 0

is identically zero, and is not equal to the function except at

x = 0

. Fortunately, we'll only be working with polynomials and powerseries, so we won't need to worry at all about pathological functions like this. In fact, we won't even really have to worry about convergence of the power series we deal with! , then

g

is represented near

a

by its Taylor series centered at

a

g (x) = g (a) + g^{'} (a) (x - a) + \frac{g^{''} (a)}{2!} {(x - a)}^{2} + \frac{g^{(3)} (a)}{3!} {(x - a)}^{3} + ...

and the behavior of $g$ very close to $a$ is completely determined (in a sense that we will make more precise in the future) by the first non-constant term of the Taylor series.

In studying functions of several variables, when ${\nabla f |}_{p} = 0$ , it also makes sense to look at higher derivatives of $f$ at $p$ . In fact, Taylor series work fine in several variables. The idea is the same as it is in the one variable case: we find a polynomial of degree $n$ in several variables all of whose partial derivatives up to order $n$ agree with those of the function $f$ . Letting $n \to \infty$ , we get an infinite power series representation in several variables for $f (x, y)$ centered at $p = (x_{0}, y_{0})$ that looks like:

\begin{matrix} f (x, y) & = f (p) + {(\frac{\partial f}{\partial x}|}_{p} (x - x_{0}) + {(\frac{\partial f}{\partial y}|}_{p} (y - y_{0})] \\ + \frac{1}{2!} \cdot {(\frac{\partial^{2} f}{\partial x^{2}}|}_{p} {(x - x_{0})}^{2} + \frac{2}{2!} \cdot {(\frac{\partial^{2} f}{\partial x \partial y}|}_{p} (x - x_{0}) (y - y_{0}) + \frac{1}{2!} \cdot {(\frac{\partial^{2} f}{\partial y^{2}}|}_{p} {(y - y_{0})}^{2} \\ + \frac{1}{3!} \cdot {(\frac{\partial^{3} f}{\partial x^{3}}|}_{p} {(x - x_{0})}^{3} + \frac{3}{3!} \cdot {(\frac{\partial^{3} f}{\partial x^{2} \partial y}|}_{p} {(x - x_{0})}^{2} (y - y_{0}) + \frac{3}{3!} \cdot {(\frac{\partial^{3} f}{\partial x \partial y^{2}}|}_{p} (x - x_{0}) {(y - y_{0})}^{2} \\ + \frac{1}{3!} \cdot {(\frac{\partial^{3} f}{\partial y^{3}}|}_{p} {(y - y_{0})}^{3} + ... \\ = \sum_{n = 0}^{\infty} \frac{1}{n!} \sum_{k = 0}^{n} (\binom{n}{k}) {(\frac{\partial^{n} f}{\partial x^{k} \partial y^{n - k}}|}_{p} {(x - x_{0})}^{k} {(y - y_{0})}^{n - k} . \end{matrix}

If $f$ is a polynomial to start with, the resulting “Taylor series” will have only finitely many non-zero terms (Why?). For example, if we expand $f (x, y) = y^{2} - x^{3} + 12 x - 16$ around the singular point $(2, 0)$ , we get

f (x, y) = y^{2} - 6 {(x - 2)}^{2} - {(x - 2)}^{3} .

A computer algebra system can compute these Taylor series expansions for us. For example, the Sage command

x,y = var("x y"); taylor(x^2*y + x*y^2, (x,3), (y,-1),10)

produces the output Really, I used the Sage command: latex(taylor(x2*y + x*y2, (x,3), (y,-1),10)) to produce output I could copy directly into a .tex file.

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Source: OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34

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