arXiv subjects

Zhan Jiang

Publications and source records attributed to Zhan Jiang.

5 recordsLinked to original sources

arXiv2023

Rationality for arbitrary closure operations and the test ideal of full extended plus closure

We extend the notion of F-rationality to other closure operations, inspired by the work of Smith, Epstein and Schwede, and Ma and Schwede, which describe F-rationality in terms of the canonical module and top local cohomology module. We give conditions for a closure operation cl on a Cohen-Macaulay complete local ring under which cl-rationality is equivalent to parameter ideals being cl-closed. We also demonstrate that full extended plus closure as defined by Heitmann and weak full extended plus closure as defined by the first named author have no big test elements.

math.AC↗

arXiv2021

RandomPoints package for Macaulay2

We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly to identify rational and geometric points in a variety over a finite field. We provide tools to estimate the dimension of a variety. We also present methods to obtain non-vanishing minors of a given size in a given matrix, by evaluating the matrix at a point.

math.AG↗

arXiv2021

Test elements in equal characteristic semianalytic algebras

We establish a series of results showing that the Jacobian ideal is contained in the test ideal. After proving a new result in characteristic $p$ for complete rings over a field $K$, we extend this containment to equal characteristic 0 for affine, analytic, affine-analytic and semianalytic $K$-algebras. We also prove a result on equiheight algebras essentially of finite type over excellent local rings.

math.AC↗

arXiv2021

Closure operations in complete local rings of mixed characteristic

Extended plus (epf) closure and rank 1 (r1f) closure are two closure operations introduced by Raymond C. Heitmann for rings of mixed characteristic. Recently, he and Linquan Ma proved that epf closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that epf closure satisfies what we call the $p$-colon-capturing property. Based on that, we define a new closure notion called "weak epf closure", and prove that it satisfies the generalized colon-capturing property and some other colon-capturing properties. This gives a new proof of the existence of big Cohen-Macaulay algebras in the mixed characteristic case. We also show that any module-finite extension of a complete local domain is epf-phantom, which generalizes a result of Mel Hochster and Craig Huneke about "phantom extensions". Finally, we prove some related results in characteristic $p$.

math.AC↗

arXiv2020

Functional equations in polynomials

We determine all F,G in C[X] of degree at least 2 for which the semigroup generated by F and G under composition is not the free semigroup on the letters F and G. We also solve the same problem for F,G in X^2 C[[X]], and prove partial analogues over arbitrary fields.

math.DS↗