I promised to continue with a constructive proof of the main theorem of Algebra. However, due to the restrictions of this blog, we will get only an outline of the proof. The details „are left to the reader“.

Given any polynomial p of degree n, and an epsilon, we have to find an effective algorithm to find a point z in the complex plane such that

\(|p(z)| \le \epsilon \)

First of all, a polynomial p is equi-continous on any bounded set. We can effectively compute the modulus of continuity by writing

\(p(z+h) – p(z) = h q(z,h) \)

with a polynomial q in z and h. We only have to estimate q for modulus z and h less than R. In fact, we get

\(|p(z+h) – p(z)| \le C |h| \)

where C depens on R.

The next ingredient is the winding number of a path in the complex plane. If

\(\gamma : [0,1] \to C \)

is a closed path, bound below by some c>0, and this mapping is equi-continuous, we can associate a winding number to the path. For this, we choose d>0 such that

\(|t-s|<d \rightarrow |\gamma(t)-\gamma(s)| < c \)

Then we choose points

\(0=t_0 < t_1 < \ldots < t_n = 1 \)

spacing less than d, and add the angles

\(\angle \gamma(t_i)-0-\gamma(t_{i+1}) \)

These angles will always be smaller than pi. The sum will be a multiple of 2pi and this is the winding number of the path. It is now possible to see, that adding further points does not change the winding number. So the winding number will not depend on the chosen points.

Note that we can effectively compute a lower bound for the modulus of the path (and also an upper bound), if we know the modulus of continuity of the path function.

Once we have that, we prove that for a large enough R the path

\(p(R e^{it}) \)

has winding number n, where n is the degree of p. This is true since p is essentially z to the power n for large z.

So the image of the boundary of a circle under p runs around 0 n times. The same is true for a large rectangle. However this does not matter. We can divide this large set in finitely many sets with half diameter. The boundaries of these sets define paths. We have to check that sum of the winding numbers of these paths must add to n, the winding number of the boundary of the complete set. This is not difficult to see, since the divisions will be passed back and forth. So one of the divisions must have a positive winding number.

Continuing we get sets with smaller and smaller diameter. We continue this, until either the diameter is less than a given bound, or the path comes closer to 0 than a given bound. Note that we can effectively check with finitely many evaluations, if the path comes close enough to 0 or not.

That finishes the outline of a constructive, effective proof of the main theorem of Algebra. Of course, this all is in the spirit of Brower, who used this idea to prove Brower’s fixed point theorem.