Rational, Replacement, and Local Invariants of a Group Action

Freeness and invariants of rational plane curves

Given a parameterization $ϕ$ of a rational plane curve C, we study some invariants of C via $ϕ$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $ϕ$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $ϕ$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational c…

两量子比特对称混合态的有理局域酉不变量

Rational Local Unitary Invariants of Symmetrically Mixed States of Two Qubits

本文研究两量子比特对称混合态的有理局域酉不变量，证明其不变量域为纯超越的，并构造出总是有理的显式几何商。

F-rationality of the ring of modular invariants

Using the description of the Frobenius limit of modules over the ring of invariants under an action of a finite group on a polynomial ring over a field of characteristic $p>0$ developed by Symonds and the author, we give a characterization of the ring of invariants with a positive dual $F$-signature. Combining this result and Kemper's result on depths of the ring of invariants under an action of a permutation group, we give an example of an $F$-rational, but non-$F$-regular ring of invariants under the action of a finite group.

有限阿贝尔群不变域有理生成元的次数界

Degree bounds for rational generators of invariant fields of finite abelian groups

本文将格论方法从多项式生成元推广到有理生成元，证明有限阿贝尔群有理不变量域的已有次数界在稍作修改后仍然成立。

A Matrix Approach to the Rational Invariants of Certain Classical Groups over Finite Fields of Characteristic Two

Let ${\mathbb F}_q$ be a finite field of characteristic two and ${\mathbb F}_q(X_1,...,X_n)$ a rational function field. We use matrix methods to obtain explicit transcendental bases of the invariant subfields of orthogonal groups and pseudo-symplectic groups on ${\mathbb F}_q(X_1,...,X_n)$ over ${\mathbb F}_q$.

有理幂、不变理想与求和公式

Rational powers, invariant ideals, and the summation formula

本文利用多面体和格点描述了几类不变理想的 rational powers 和 Rees valuations，证明了 Mustaţă-Takagi 求和公式对这些理想成立，并对 normal domains 中任意理想证明了其在充分大有理数下成立的较弱版本。