Narrow Results

In this paper, we study the generators of the module of derivations of the ring of invariants for some dihedral groups. Let $k$ be an algebraically closed field of characteristic $0$ and $D_{n,q}(\subset \mathrm{\text{GL}}(2,k))$ be a finite dihedral group. Let $R=k{[X,Y]}^{D_{n,q}}$ be the ring of invariants obtained by the linear action of $D_{n,q}$ on $k[X,Y]$. In [ R. V. Gurjar and A. Patra, On minimum number of generators for some derivation modules, J. Pure Appl. Algebra$226$(11) (2022) 107105], Gurjar–Patra proved that $\mu ({\text{Der}}_k(R))<\left|D_{n,q}\right|$. We will give a better bound on $\mu ({\text{Der}}_k(R))$ in this paper.

This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.

Let $R,T$ be commutative rings with identity such that $R\subseteq T$. We recall that $R\subseteq T$ is called a $\lambda$-extension of rings if the set of all subrings of $T$ containing $R$ (the “intermediate rings”) is linearly ordered under inclusion. In this paper, a characterization of integrally closed $\lambda$-extension of rings is given. For example, we show that if $R$ is a local ring, then $R\subseteq T$ is an integrally closed $\lambda$-extension of rings if and only if there exists $𝔮\in \mathrm{\text{Spec}}(R)$ such that $T=R_{𝔮},𝔮=T𝔮$ and $R/𝔮$ is a valuation domain. Let $R$ be a subring of $T$ such that $R$ is invariant under action by $G$, where $G$ is a subgroup of the automorphism group of $T$. If $R\subseteq T$ is a $\lambda$-extension of rings, then $R^G\subseteq T^G$ is a $\lambda$-extension of rings under some conditions.

Let $R,T$ be commutative rings with identity such that $R\subseteq T$. A ring extension $R\subseteq T$ is called a $\mathrm{\Delta }$-extension of rings if $R_1+R_2$ is a subring of $T$ for each pair of subrings $R_1,R_2$ of $T$ containing $R$. In this paper, a characterization of integrally closed $\mathrm{\Delta }$-extension of rings is given. The equivalence of $\mathrm{\Delta }$-extension of rings and $\lambda$-extension of rings is established for an integrally closed extension of a local ring. Over a finite dimensional, integrally closed extension of local rings, the equivalence of $\mathrm{\Delta }$-extensions of rings, FIP, and FCP is shown. Let $R$ be a subring of $T$ such that $R$ is invariant under action by $G$, where $G$ is a subgroup of the automorphism group of $T$. If $R\subseteq T$ is a $\mathrm{\Delta }$-extension of rings, then $R^G\subseteq T^G$ is a $\mathrm{\Delta }$-extension of rings under some conditions. Many such $G$-invariant properties are also discussed.

In this paper, we study the derivation module of the ring of invariants of $k[X,Y]$ under the linear action of dihedral groups $D_{n,q}<GL(2,k)$ mentioned in a paper by Riemenschneider [Die Invarianten der endlichen Untergruppen von $GL(2,ℂ)$, Math. Zeitsch. 153 (1977) 37–50]. We obtained an explicit generating set for the derivation module of $R=k{[X,Y]}^{D_{n,q}}$. We show that $\mu ({\mathrm{\text{Der}}}_k(R))\le O(D_{n,q})$.

We use an algorithm based on the notion of transvectants from classical invariant theory in determining the form of Stanley decomposition of the ring of invariants for the coupled Takens–Bogdanov systems when the Stanley decompositions of the Jordan blocks of the linear part are known at each stage. The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module.