# Maximum number of critical points for multivalued polynomial

• Jun 10th 2013, 10:00 PM
gralla55
Maximum number of critical points for multivalued polynomial
For a function of a single variable, the maximum number of critical points is just equal to the order of the polynomial -1. So a fourth degree polynomial function has a maximum of 3 critical points.

I was wondering what the case would be for a function of n variables. Let's say you have a function f(x,y) = x^3 + y^3 + ..... etc

I have a feeling the maximum number of critical points should be 4, or just the highest powers of x and y -1 added together. I don't know if it's right, or how to prove it though...
• Jun 11th 2013, 12:57 AM
chiro
Re: Maximum number of critical points for multivalued polynomial
Hey gralla55.

You have to multiply the powers of each independent variable. So if you have x^3 and y^3 then the maximum number should be (3-1)*(3-1) = 2*2 = 4.

If one variable is dependent on another then transform them to the same variable and look at the highest degree and subtract 1.
• Jun 11th 2013, 08:09 AM
HallsofIvy
Re: Maximum number of critical points for multivalued polynomial
Every time you say "function", you mean "polynomial", right?