On Computation of Kolchin Characteristic Sets: Ordinary and Partial Cases

Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation

We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate Kolchin-type dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. In particular, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. As a consequence, it is shown that the…

A bound for a typical differential dimension of system of linear differential equations

We prove upper and lower bounds for leading coefficient of Kolchin dimension polynomial of systems of partial linear differential equations in codimension two.

Ordinary and symbolic powers are Golod

Let $S$ be a positively graded polynomial ring over a field of characteristic 0, and $I\subset S$ a proper graded ideal. In this note it is shown that $S/I$ is Golod if $\partial(I)^2\subset I$. Here $\partial(I)$ denotes the ideal generated by all the partial derivatives of elements of $I$. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals.

Completeness in Partial Differential Fields

We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and use it to give a new class of examples of complete partial differential varieties. We use methods from both differential algebra and model theory.

边理想的普通幂与符号幂在正则性和深度方面的比较

Comparisons between ordinary and symbolic powers of edge ideals with respect to regularity and depth

本文研究边理想普通幂与符号幂在正则性和深度上的差异，证明单纯图情形下二者正则性一致，并给出深度不等式成立条件及差异模的Krull维数图论公式。

D-代数函数的算术

Arithmetic of D-Algebraic Functions

本文研究D-代数函数解的有理表达式所满足的代数微分方程，提出单变量最小阶ADE推导方法及多变量通用算法，并给出阶数界限，应用于物理、统计和符号积分。