We prove that a number field $K$ satisfies the following property (B) if and only if the ray class group of $K$ defined modulo 4 is trivial. (B): For any tame abelian extensions $N_1$ and $N_2$ over $K$ of exponent 2, the composite $N_1N_2/K$ has a relative normal integral basis (NIB) if both $N_1/K$ and $N_2/K$ have a NIB.

In this article, we construct generic polynomials over $\boldsymbol{Q}$ with two parameters for all transitive subgroups of the symmetric group of degree 5 by considering the action on the moduli space of the projective line with ordered five marked points. Although polynomials having such properties are already known, our device is unifying through all the cases, and in some cases we obtain polynomials with much simpler coefficients.

Given $p$ polynomials with coefficients in a commutative unitary integral ring $\mathcal{C}$ containing $\mathbf{Q}$, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme $V \subset \operatorname{Spec}(\mathcal{C})$. We prove the existence of such a non zero rational polynomial which covers and generalizes previous existing results by H. Biosca. When $\mathcal{C}$ is the ring of an algebraic or analytic space, we deduce a stratification of the space of the parameters such that on each stratum, there is a non zero rational polynomial which is a Bernstein-Sato polynomial for any point of the stratum. This generalizes a result of A. Leykin obtained in the case $p = 1$.

In this paper we show that any convex quasi-homogeneous projective domain whose boundary is everywhere twice differentiable except possibly at a finite number of points is homogeneous.

In this paper, we shall prove the existence of a $t$-conformal measure supported on conical points for a rational map $T$ whose critical points in its Julia set are recurrent. In particular, there always exists a $t$-conformal measure supported on conical points for the polynomial $T(z) = z^d + c$.

We study the unique continuation property of two-dimensional Dirac equations with Aharonov-Bohm fields. Some results for the unperturbed Dirac operator are given by De Carli-\={O}kaji [2]. We are interested in the problem how the singularity of Aharonov-Bohm fields at the origin influences the unique continuation property.