The following proposition shows how roots of polynomials and their derivatives are somewhat attracted to each other in a rough sense. They exhibit some sort of clustering at the ends of intervals of reasonable logarithmic length.
Proposition 1 Let be a real polynomial of degree s.t. , and call , for its complex roots – which we assume are all distinct – ordered by increasing modulus ( thus). Let be big (with constant depending only on the degree of ) and suppose there exists a index for which
- when one has
- there exists a constant depending only on the degree of such that on the same interval as in the previous point it holds
The first conclusion of the theorem is telling us that in between – but far enough from – well separated roots ( and are separated by a factor of at least ), , i.e. the polynomial behaves essentially like a monomial , whose degree can be less that the degree of . The second conclusion is telling us that on the interval , that is , so that , and therefore , is either strictly increasing or strictly decreasing. In particular is strictly increasing on and strictly decreasing on .
Proof: As for the first point, write and observe that when the term is essentially because the roots are smaller than that by a huge factor, and when the opposite happens, i.e. . To be more precise, when one has and thus
while if it is and thus
Taking the product of all factors one recognizes the upper and lower terms above.
Now, since one has
Before going further, a little heuristics about the sum. The terms in the sum for are now of order of magnitude , while the terms for have order of magnitude . Since is a huge constant (and we can set how big it is) one should expect that the first terms dominate on the last , which of course it is. So, assume wlog that ,
and each term of the second sum contributes with at most
while the first term being a modulus is certainly bigger or equal that its real part, and then for each term
so in the end the logarithmic derivative is bounded from below by
and the constant can be made positive – bigger than a constant – for every by taking big enough (the first term in inside the brackets is and the second one is ).
Notice that if then the logarithmic derivative is still bounded from below, and that tells us is monotonic on , since is a simple root (thus is monotonic through ). On the other hand if then , as it is to be expected, and moreover the logarithmic derivative is again bounded from below, so that is increasing on and decreasing on .
One last comment about the structure that the proposition yields us. Between intervals of the form there are intervals of the form that contain at least a root and where the polynomial doesn’t behave so nicely. We can call them Type 1 intervals, and call Type 2 intervals the others. This leads to a partition of the positive reals (that extends to all reals by taking the symmetric intervals, reflected with respect to 0) into Type 1 and Type 2 intervals, given by
where some (or all, if the roots are concentrated around ) of the intervals might be empty (we adopt the convention that if then is a proper interval but ). They are not empty when
but this doesn’t prevent from having a very small measure, so in some applications it might be better to assume that Type 2 intervals have the property
so that the measure of is bounded from below by , and all the other intervals are of Type 1 instead.