User:Dom walden/Multivariate Analytic Combinatorics/Smooth Point via Residue Forms
Introduction edit
Here, we prove a theorem very similar to that of the previous chapter.
The proof is shorter but involves more advanced mathematics. It has the advantage that it does not assume the same minimality hypotheses.
Theorem edit
If with a simple pole on . is smooth above height . If there is a subset of nondegenerate critical points on at height such that in . Then, there is a compact neighbourhood of ... Then[1]
Proof edit
Let and be two components of where is not bounded from below on . Let for some and for some .[2]
The intersection class is represented by the intersection of and a homotopy between and which intersects transversely.[3]
If we choose the homotopy such that its time cross-sections are tori that expand with and go through , perturbed to intersect transversely, then the class can be represented by a smooth (d - 1)-chain on on which the height reaches its maximum at .[4]
By the Cauchy coefficient formula and residue theorem:
As a result of ...[5]
By theorem 5.3 and the change of variables gives[6]
Definitions edit
Intersection class edit
Transverse edit
Example edit
Notes edit
References edit
- Pemantle, Robin; Wilson, Mark C. (2013). Analytic Combinatorics in Several Variables (PDF) (1st ed.). Cambridge University Press.
- Pemantle, Robin; Wilson, Mark C.; Melczer, Stephen (2024). Analytic Combinatorics in Several Variables (PDF) (2nd ed.). Cambridge University Press.