Proof. The first assertion is Exercise III.5.2 of [1]. The degree assertion can be deduced by modifying the proof of the same exercise to remove the first term of the exact sequence, using Corollary 4.1 of [2] (and noting that invariance of cohomology under field extension means we can reduce immediately to the case of an infinite field). The last equality follows from Serre’s theorem on vanish...

The appellation “Hilbert” is sometimes reserved for cases where D is a fundamental discriminant (in which case HD(X) is more generally called a ring class polynomial), but we shall use the term Hilbert class polynomial to refer to HD(X) in general. Our first objective is to use the fact that ΦN ∈ Z[X,Y ] to prove that HD ∈ Z[X]. We require the following lemma. Lemma 22.2. If N is prime then the...

H(n) = uniform bound for the number of limit cycles of (1) . One way to formulate the Hilbert 16th problem is the following: Hilbert 16th Problem (HP). Estimate H(n) for any n ∈ Z+. To prove that H(1) = 0 is an exercise, but to find H(2) is already a difficult unsolved problem (see [DRR,DMR] for work in this direction). Below we discuss two of the most significant branches of research HP has ge...