The Action of a Plane Singular Holomorphic Flow on a Non-invariant Branch

We study the dynamics of a singular holomorphic vector field at $(\mathbb{C}^{2},0)$. Using the associated flow and its pullback to the blow-up manifold, we provide invariants relating the vector field, a non-invariant analytic branch of curve, and the deformation of this branch by the flow. This leads us to study the conjugacy classes of singular branches under the action of holomorphic flows. In particular, we show that there exists an analytic class that is not complete, meaning that there are two elements of the class that are not analytically conjugated by a local biholomorphism embedded in a one-parameter flow. Our techniques are new and offer an approach dual to the one used classically to study singularities of holomorphic vector fields.

On the Borel mapping in the quasianalytic setting

The Borel mapping takes germs at $0$ of smooth functions to the sequence of iterated partial derivatives at $0$. We prove that the Borel mapping restricted to the germs of any quasianalytic ultradifferentiable class strictly larger than the real analytic class is never onto the corresponding sequence space.

An application of the -adic analytic class number formula

We propose an algorithm to verify the$p$-part of the class number for a number field$K$, provided$K$is totally real and an abelian extension of the rational field$\mathbb{Q}$, and$p$is any prime. On fields of degree 4 or higher, this algorithm has been shown heuristically to be faster than classical algorithms that compute the entire class number, with improvement increasing with larger field degrees.

A quasianalytic singular spectrum with respect to the Denjoy-Carleman class

Making use of the FBI (Fourier-Bros-Iagolnitzer) transforms we simplify the quasianalytic singular spectrum for the Fourier hyperfunctions, which was defined for distributions by Hörmander as follows; for any Fourier hyperfunction u, (x0, ξ0) does not belong to the quasianalytic singular spectrum W FM(u) if and only if there exist positive constants C, γ and N, and a neighborhood of x0 and a conic neighborhood Г of ξ0 such thatfor all x ∈ U, |ξ| ∈ Γ and |ξ| ≥ N, where M(t) is the associated function of the defining sequence Mp. This result simplifies Hörmander’s definition and unify the singular spectra for the C∞ class, the analytic class and the Denjoy-Carleman class, both quasianalytic and nonquasianalytic.

On the class number of the lpth cyclotomic number field

The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.