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This is a collection of lecture notes and problem sets from MATH 499: VIGRE Computational Algebraic Geometry at Rice University in 2010.
A point is a singular point of a plane curve if and only if so that is on the curve, and
In single variable calculus, for studying points where the first derivative of a function
is zero, it is often helpful to study the higher derivatives. For example, if
and
, then
has a local minimum at
. More generally, if
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
You can check that this function is infinitely differentiable, but
so fast as
that
for all
. Thus its Taylor series centered at
is identically zero, and is not equal to the function except at
.
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
is represented near
by its Taylor series centered at
,
and the behavior of very close to 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 , it also makes sense to look at higher derivatives of at . 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 in several variables all of whose partial derivatives up to order agree with those of the function . Letting , we get an infinite power series representation in several variables for centered at that looks like:
If is a polynomial to start with, the resulting “Taylor series” will have only finitely many non-zero terms (Why?). For example, if we expand around the singular point , we get
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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