A new family of ideals with the doubly exponential ideal membership property

关于一个理想隶属问题

On an ideal membership problem

本文研究正则环中理想隶属问题，证明指数最优性，并针对特征零域上齐次多项式给出更强结果。

Generic bounds for Frobenius closure and tight closure

We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P=k[x_0,..., x_d], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d+1. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a generic bound for membership in the Frobenius closure. If d is at most 2, then the bound for ideal membership in P can be computed from …

Degree Bounds for Gröbner Bases in Algebras of Solvable Type

We establish doubly-exponential degree bounds for Gröbner bases in certain algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning). The class of algebras considered here includes commutative polynomial rings, Weyl algebras, and universal enveloping algebras of finite-dimensional Lie algebras. For the computation of these bounds, we adapt a method due to Dubé based on a generalization of Stanley decompositions. Our bounds yield doubly-exponential degree bounds for ideal membership and syzygies, generalizing the classical results of Hermann and Seidenberg (in the c…

On the Complexity of CSP-based Ideal Membership Problems

In this paper we consider the Ideal Membership Problem (IMP for short), in which we are given real polynomials $f_0,f_1,\dots, f_k$ and the question is to decide whether $f_0$ belongs to the ideal generated by $f_1,\dots,f_k$. In the more stringent version the task is also to find a proof of this fact. The IMP underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. In the majority of such applications the IMP involves so called combinatorial ideals that arise from a variety of discrete combinatorial problems. This restriction makes th…

Regularity of primes associated with polynomial parametrisations

We prove a doubly exponential bound for the Castelnuovo-Mumford regularity of prime ideals defining varieties with polynomial parametrisation.

等变 Gröbner 基的可计算性

Computability of Equivariant Gröbner bases

本文证明了在满足 Hilbert 基性质时等变理想 Gröbner 基可计算且成员问题可判定，并给出了该问题不可判定的充分条件。

理想成员问题与阿贝尔群

The Ideal Membership Problem and Abelian Groups

本文证明，对于仿射约束语言，理想成员问题可以在多项式时间内解决，前提是输入多项式次数有界。

素理想与具有大正则性的三生成理想

Prime ideals and three-generated ideals with large regularity

本文构造了例子，证明三生成理想和素理想的正则性下界至少为双重指数，显著改进了此前超多项式的下界。

Segre-Driven Radicality Testing

We present a probabilistic algorithm to test if a homogeneous polynomial ideal $I$ defining a scheme $X$ in $\mathbb{P}^n$ is radical using Segre classes and other geometric notions from intersection theory. Its worst case complexity depends on the geometry of $X$. If the scheme $X$ has reduced isolated primary components and no embedded components supported the singular locus of $X_{\rm red}=V(\sqrt{I})$, then the worst case complexity is doubly exponential in $n$; in all the other cases the complexity is singly exponential. The realm of the ideals for which our radical testing procedure requ…

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely al…