Sturm's theorem

Sturm's theorem is a math rule that tells us how many roots a polynomial has, which are the numbers that make the polynomial equal to zero. It's like how many fingers you have on your hand – you can count them, right? Well, Sturm's theorem helps us count the roots of a polynomial.

Imagine you have a bunch of numbers written down in order, such as 1, 3, 5, 7, and 9. These numbers form a sequence. Now, let's say you want to know how many numbers in this sequence are between 2 and 8. To figure this out, you could compare each number in the sequence to the number 2 and the number 8, and count the amount of times the sequence "crosses" between 2 and 8.

Here's where it gets a bit trickier: Sturm's theorem does something similar to this sequence, but instead of looking at how many numbers are between two specific numbers, it looks at how many "sign changes" happen between two specific points in the sequence. A "sign change" happens when a positive number becomes negative or a negative number becomes positive.

Now, back to the polynomials. Imagine you have a polynomial equation like x^2 - 2x + 1 = 0. This means you're trying to find out when the polynomial equals zero. You could try plugging in a bunch of numbers into this equation, but that would take forever. Instead, you can use Sturm's theorem to figure out how many roots (numbers that make the equation equal zero) there are.

Sturm's theorem looks at the polynomial and creates a sequence of smaller polynomials based on it. Then, it counts the number of sign changes in the sequence between two specific points. The number of sign changes gives you the number of roots the original polynomial has.

So, for the polynomial x^2 - 2x + 1 = 0, Sturm's theorem creates a sequence of polynomials:

x^2 - 2x + 1
x^2 - 2x
-x + 1

Now, you might notice that the first two polynomials look pretty similar. In fact, you can subtract the second polynomial from the first and get x - 1. This is the third polynomial in the sequence.

So, let's look at the sign changes between two important points in this sequence (in this case, -infinity and infinity):

Between -infinity and infinity: 1 sign change
Between -1 and infinity: 0 sign changes

This means that the original polynomial (x^2 - 2x + 1 = 0) has one root. To find out what that root is, you can solve the equation (in this case, it's x = 1).

Overall, Sturm's theorem helps us quickly figure out how many roots a polynomial has by looking at the number of sign changes in a sequence of smaller polynomials.