Graphing Multivariable Polynomials

Date: 02/20/2005 at 12:30:55
From: Eric
Subject: Multivariable Polynomials

I have a few questions about multivariable polynomials.  Say we have
the polynomial x + y + z, can this be graphed?  I also really don't
know anything about the z axis or how to graph on it. 

I guess I'm really just looking for more information about 
multivariable polynomials.

Thanks.

Date: 02/20/2005 at 14:27:12
From: Doctor Tom
Subject: Re: Multivariable Polynomials

Hi Eric,

We live in a three-dimensional world, so it's tough to "graph" 
polynomials with more than two variables.  Let's go back to 
single-variable polynomials for a second.  When you graph f(x) = x^2 -
3x + 2, you usually graph the input, x, on one axis and the output,
f(x), on the other.  You need two dimensional paper to do this.

Now consider something like this:  f(x, y) = x^2 - xy + y - 6.  For
each pair of values you choose to be x and y, there is a single 
output.  This can be "graphed" as follows:  Draw an x-axis and a
y-axis on a sheet of paper and place that paper on a table.  Now, each
point on the paper corresponds to an x and y pair, so calculate 
f(x, y) for that pair.  The output is a number, and (somehow) put a
dot in space that distance above the paper.  If the output is
negative, put the dot below the paper.  If you do this (conceptually)
for every (x, y) pair, you'll get a surface whose height above or
below the paper represents the output value.

Let's look at a simpler example: f(x, y) = x + y - 5.  The "surface"
in this case will be a plane that slices through your paper along the
line x + y - 5 = 0.  Every point along this line is a solution--there
are, as you noticed, an infinite number of them.

With curved surfaces, the intersection with the paper may be curved
lines or combinations of curved lines.

Unfortuantely, it's tough to go to the next stage.  For something like
your example of f(x, y, z) = x + y + z, you need a three-dimensional
volume just to represent the inputs.  Since we only live in a
three-dimensional world, there's no fourth direction you can go to
graph the output.

There are tricks you can do that might help visualize what's going on.
Imagine that the output happened to be a number between 0 and 1 in
all cases.  You could say to yourself: "I'll represent 0 by red and 1
by violet, and the numbers in between by the colors of the rainbow
going from red to violet.  Then you could have a 3-D grid where the
output of each triple is represented by the appropriate color as a
series of dots in the grid.  You can't fill in all the dots or you
can't see inside, but you could space them, and imagine looking at the
whole mess on a computer screen where you could change your viewpoint.

The problem of "graphing" high-dimensional systems is very important,
and there's a lot of work in this area of visualization.  Edward Tufte
has a wonderful series of books on visualization of these sorts of
things and many others.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/