In our previous paper on Briançon–Speder Polynomials, we gave a classification of the connected components of $\mu^{*}$-constant strata on the space of weighted homogeneous polynomials of three variables which generalize Briançon and Speder type polynomials. In this paper, we give further remarks on the $\mu^{*}$-constant strata of higher dimensional weighted homogeneous polynomials.
We find new bi-Lipschitz invariants for holomorphic functions of two complex variables. Our new invariants can distinguish function germs that are otherwise indistinguishable by previous methods, as illustrated by the examples given in Section 4.
Oka introduced the concept of mixed polynomials and started to investigate how a study for singularities of mixed polynomials similar to the study of singularities of polynomials is possible. We introduce a mixed toric modification as a mixed analogy of toric modifications and discuss when this provides an analogy of resolutions of singularities defined by mixed polynomials. A mixed toric modification is associated with a mixed fan, which is a notion we introduce in the paper. They provide several combinatorial data for singularities of mixed polynomials. We define the notion of mixed Newton non-degeneracy for mixed polynomials and show that a mixed toric modification provides a semi-algebraic or real algebraic analogue of resolutions of singularities under mixed Newton non-degeneracy condition. Our approach allows us a combinatorial description of the topology of singularities of mixed polynomials, which are mixed Newton non-degenerate, and we show formulas for the Euler characteristics and the monodromy zeta function of nearby fibers. We also show how the dual graphs of analogy of resolution of singularities of such mixed polynomials are obtained in low dimensions.
We prove the Zariski density of certain set of CM points lying in self-products of elliptic modular curves, considered by Hida. A new ingredient in our argument is a finiteness result on irreducible components for the inverse images in infinite coverings of self-products of modular curves.
In this paper we study compact Riemannian symmetric spaces whose maximal tori are as simple as possible: metric products of circles. In particular, using representation theory, we construct explicitly an isometric embedding of such spaces $X$ into Euclidean space with the property that geodesic symmetries of $X$ extend to reflections along normal spaces. The converse is also true: extrinsically symmetric submanifolds have rectangular maximal tori. Further, such embedding of $X$ is unique up to congruence, and all isometries of $X$ extend to isometries of the ambient Euclidean space.
