The Hilbert Zonotope and a Polynomial Time Algorithm for Universal Grobner Bases

A Tutte polynomial for toric arrangements

We introduce a multiplicity Tutte polynomial M(x,y), with applications to zonotopes and toric arrangements. We prove that M(x,y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincare' polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x,y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume …

Gröbner基、对称矩阵与C型Kazhdan-Lusztig簇

Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties

研究一般对称矩阵子式生成的多项式理想，作为C型Kazhdan-Lusztig簇的概型定义理想，给出了其Gröbner基、初始理想素分解及基于pipe dream的Hilbert级数公式。

Quadratic ideals and Rogers-Ramanujan recursions

We give an explicit recursive description of the Hilbert series and Gröbner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers-Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.

Gröbner geometry of Schubert polynomials

Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons. This results in a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. Given a permutation w in S_n, we consider a determinantal ideal I_w whose generators are certain minors in the generic n x n matrix (filled with independent variables). Using `multidegrees' as simple algebraic substitutes for torus-equivar…

Independent Sets from an Algebraic Perspective

In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen-Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P=P. Moreover, we present a family of radical zero-dimensional complete intersection ideals J_P associated to a finite poset P, for which we describe a universal Gröbner basis. This implies t…

Combinatorics and geometry of power ideals

We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A, and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of the resulting fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results due to Dahmen-Micchelli, Holtz-Ron…

On a class of power ideals

In this paper we study the class of power ideals generated by the $k^n$ forms $(x_0+ξ^{g_1}x_1+\ldots+ξ^{g_n}x_n)^{(k-1)d}$ where $ξ$ is a fixed primitive $k^{th}$-root of unity and $0\leq g_j\leq k-1$ for all $j$. For $k=2$, by using a $\mathbb{Z}_k^{n+1}$-grading on $\mathbb{C}[x_0,\ldots,x_n]$, we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for $k>2$. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the $k^n$ points $[1:ξ^{g_1}:\ldots:ξ^{g_n}]$ in $\mathbb{P}^n$. W…

关于乘积空间中多项式的结构及其普遍非笛卡尔性质

On the structure and generic non-Cartesianity of polynomials in product spaces

本文研究了复空间乘积上多项式的结构和非笛卡尔性质，证明了在固定次数下，多项式通常非笛卡尔，且通过格罗布纳基组方法可以算法判断非笛卡尔性，并将其与组合几何联系起来，得到精确的交集界和最优配置。

Short Rational Functions for Toric Algebra and Applications

We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed, this representation allows us to compute the Graver basis and the reduced Gröbner basis of the ideal $I_A$, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computation which replaces in this new encoding the usual reductions typical of the division algorithm. We describe o…

Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

The current paper deals with the enumeration and classification of the set $\mathcal{SOR}_{r,n}$ of self-orthogonal $r\times r$ partial Latin rectangles based on $n$ symbols. These combinatorial objects are identified with the independent sets of a Hamming graph and with the zeros of a radical zero-dimensional ideal of polynomials, whose reduced Gröbner basis and Hilbert series can be computed to determine explicitly the set $\mathcal{SOR}_{r,n}$. In particular, the cardinality of this set is shown for $r\leq 4$ and $n\leq 9$ and several formulas on the cardinality of $\mathcal{SOR}_{r,n}$ are…