不可分解多项式及其谱

Homological mirror symmetry of indecomposable Cohen-Macaulay modules for some degenerate cusp singularities

Burban-Drozd showed that the degenerate cusp singularities have tame Cohen-Macaulay representation type, and classified all indecomposable Cohen-Macaulay modules over them. One of their main example is the non-isolated singularity $W=xyz$. On the other hand, Abouzaid-Auroux-Efimov-Katzarkov-Orlov showed that $W=xyz$ is mirror to a pair of pants. In this paper, we investigate homological mirror symmetry of these indecomposable Cohen-Macaulay modules for $xyz$. Namely, we show that closed geodesics (with a flat $\mathbb{C}$-bundle) of a hyperbolic pair of pants have a one-to-one correspondence w…

更高深度下环面Kac多项式的正性

Positivity for toric Kac polynomials in higher depth

该研究证明更高深度下计数截断幂级数环上箭图秩1绝对不可分解表示的多项式系数非负，建立箭图表示计数与矩映射纤维射流计数的合冲恒等式，并验证了Wyss的两个渐近猜想。

交换Frobenius代数上箭图局部自由表示

Locally free representations of quivers over commutative Frobenius algebras

本文利用算术傅里叶变换研究交换Frobenius代数上箭图的局部自由表示，证明了有限域下绝对不可分解局部自由表示的同构类数量与箭图定向无关，并给出了表示有限型的完整分类及计数函数的性质。

Local cohomology with support in ideals of maximal minors

Suppose that k is a field of characteristic zero, X is an r by s matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X. We study the local cohomology modules H^i_I(R), where I is the ideal of R generated by the maximal minors of X. We identify the indices i for which these modules vanish, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r)+1, and characterize all nonzero ones as submodules of certain indecomposable injective modules. These results are consequences of more general theorems regarding linearly reductive groups acting o…

非共振超平面排列的N/D猜想

The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements

本文通过Walther的结果验证了满足非共振条件的加权超平面排列的n/d猜想，并推导出强拓扑单叶猜想。

A note on the toric Newton spectrum of a polynomial

We define the toric Newton spectrum of a polynomial and we give some applications in singularity theory, combinatorics and mirror symmetry.

曲线上向量丛的代数算法

Algebraic algorithms for vector bundles over curves

本文提出用函数域极大序上的格对表示曲线上的向量丛，并给出计算行列式、同态核像、同构、上同调、扩张及不可分解直和分解的多项式时间代数算法，同时实现为SageMath包并应用于椭圆曲线向量丛分类与代数几何码构造。

Commuting matrices and volumes of linear stacks

A conjecture by Higman asserts that the number of conjugacy classes in the unipotent group of upper triangular matrices over a finite field depends polynomially on the number of elements of the field. We will study several alternative counting problems arising from quiver representations and prove explicit formulas relating the corresponding invariants to the invariants of Higman's conjecture. To do this, we develop a general framework of linear stacks over small etale sites and study volumes of these stacks and of their substacks of absolutely indecomposable objects.

Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R = \mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machiner…

Ehrhart polynomials of polytopes and spectrum at infinity of Laurent polynomials

Gathering different results from singularity theory, geometry and combinatorics, we show that the spectrum at infinity of a tame Laurent polynomial counts lattice points in polytopes and we deduce an effective algorithm in order to compute the Ehrhart polynomial of a simplex containing the origin as an interior point.

多项式乘数谱的研究

On the multiplier spectrum of polynomials

本文研究多项式乘数谱的性质，证明了其映射在模空间上普遍单射，并探讨了Ritt移动与乘数谱的关系。