The membership problem for unmixed polynomial ideals is solvable in single exponential time

The membership probl mixed polynomial i solvable in single exp

Dickenstein, A., N. Fitchas, M. Giusti and C. Sessa, The membership problem for unmixed polynomial ideals is solvable in single exponential time, Discrete Applied Mathematics 33 (1991) 73-94. Deciding membership for polynomial ideals represents a classical problem of computational commutative algebra which is exponential space hard. This means that the usual al membership problem which are base...

The Membership Problem for Polynomial Ideals in Terms of Residue Currents

We find a relation between the vanishing of a globally defined residue current on Pn and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther’s theorem, and we also obtain a generalization of that theorem. There are also connections to effective versions of the Nullstellensatz. We also provide explicit i...

The Membership Problem for Ideals in Z[x]

There exists a feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal in Z[x] if the ideal’s minimal basis is known. However, when this is not the case there is no feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal. There already exists an effective procedure to find an ideal’s minimal basis, but it depends on solvi...

Quadratic Assignment Problem: Linearizations and Polynomial Time Solvable Cases

Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree

In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1, 2, . . . ,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97 ·n · logW ) using O(n · lognW · log k) space. This is the first single-exponential time, polynomial-sp...