We give an example of hypersurfaces with non-isolated singularities in $\mathrm{P}^4(\mathbf{C})$, whose normalizations have isolated rational quadruple points only as singularities. From Schlessinger's criterion, it follows that these isolated rational singular points are rigid under small deformations.

In this paper, we present the GNB-aided MSST algorithm for the curves over finite fields that have a Gaussian normal basis of type $t \le 2$. It is based on the MSST algorithm proposed by P. Gaudry [3] at ASIACRYPT 2002. For those fields, we combine the lifting phase of the MSST algorithm and the norm computation algorithm in [6]. So the time complexity of the MSST is reduced from $O(N^{2\mu + 0.5})$ to $O(N^{2\mu + 1/ (\mu + 1)})$ and it runs faster than any other algorithms in our case.

The function $f_{\alpha}(z) = \bigl( \{(1+z)/(1-z)\}^{\alpha} - 1 \bigr) / (2\alpha)$ with a complex constant $\alpha \ne 0$ is not univalent in the disk $U = \{|z| < 1\}$ if and only if $\alpha$ is not in the union $A$ of the closed disks $\{|z + 1| \leqslant 1\}$ and $\{|z - 1| \leqslant 1\}$. By making use of a geometric quantity we can describe how $f_{\alpha}$ ``continuously tends to be'' univalent in the whole $U$ as $\alpha$ tends to each boundary point of $A$ from outside.

In this paper, we compute $L_p(1, \chi) \bmod{p}$ when $\chi$ is the nontrivial character of a real quadratic field. As a result, we give a sufficient condition for Iwasawa invariants $\mu_p(k)$, $\lambda_p(k)$ to vanish when $p$ splits in a real quadratic field $k$.

We express normalized double sine functions of integer periods $(N_1, N_2)$ via the standard double sine function of period $(1,1)$. As an application we give an Euler product expression using the di-logarithm for the double zeta function $\zeta(s, \mathbf{F}_{p^{N_1}}) \otimes \zeta(s, \mathbf{F}_{p^{N_2}})$ for a prime number $p$ and integers $N_1$, $N_2$.