Applying Rouché Theorem to Bigger Polynomials

[Mappe]

I have a problem with applying the rouché theorem to bigger polynomials. Generally,

(Az^4 + Bz^3 + Cz^2 + Dz + E) on some annulus k1 < |z-z0| < k2

So, I've tried applying the theorem by the version |f - g| < |f|, and I've chosen g as different terms in the polynomial, starting from z^4, and working down until the inequity is satisfied.

Basically what I can't figure out, is how to use the triangle inequity to evaluate the function on the contour. Also, doing the triangle-inequity-approximation, if I find it to be 3 zeros in the bigger circle, how can I know that the inequity approximation didn't ignore one of the zeros, and that there actually is 4 zeroes there? I mean, the |f - g| < |f| inequity may be satisfied even if the triangle equity says it aint? Or not?

 

[lippyka]

The Rouché theorem can be applied to polynomials of any degree, as long as the coefficients are known. The theorem states that if two functions f and g are analytic on a simply connected region, and if |f(z) - g(z)| < |f(z)| on the boundary of the region, then f and g have the same number of zeros in the region. To apply the theorem to polynomials, we first need to determine what the function g should be. Suppose we have the polynomial Az^4 + Bz^3 + Cz^2 + Dz + E. We can choose g to be the sum of all terms up to the term with the highest power of z, in this case z^4. This means g = Az^4, and therefore f-g = Bz^3 + Cz^2 + Dz + E. To determine whether the inequality |f(z) - g(z)| < |f(z)| is satisfied on the boundary of the given annulus k1 < |z-z0| < k2, we can use the triangle inequality. The triangle inequality states that for three complex numbers u, v, and w, |u+v+w| ≤ |u| + |v| + |w|. We can apply this to the inequality f-g by breaking up the sum into its individual terms. For example, |Bz^3 + Cz^2 + Dz + E| ≤ |Bz^3| + |Cz^2| + |Dz| + |E|. Now that we have an expression for the left side of the triangle inequality, we can evaluate it at each point on the boundary of the annulus and check whether it is less than |f(z)|. If it is, then the Rouché theorem tells us that f and g have the same number of zeros in the region. To answer your question about how to know that the triangle inequality approximation didn't ignore one of the zeros, the best way is to actually calculate the number of zeros in the region. This can be done by counting the number of roots of the polynomial (using something like the Sturm sequence algorithm), or by using the argument principle.