Effective Nullstellensatz and Geometric Degree for Zero-dimensional Ideals

Tropical Varieties, Ideals and an Algebraic Nullstellensatz

The objective of this paper is to introduce the fundamental algebro-geometric constructions over the extended tropical semi-ring. The study of tropical varieties, co-varieties and ideals over this extension eventually yields the theorem of the weak tropical Nullstellensatz and gives an algebraic interpretation of the tropical Nullstellensatz.

ar X iv : m at h / 05 11 05 9 v 2 [ m at h . A C ] 8 M ay 2 00 6 Tropical Varieties , Ideals and An Algebraic Nullstellensatz

The objective of this paper is to introduce the fundamental algebro-geometric constructions over the extended tropical semi-ring. The study of tropical varieties, co-varieties and ideals over this extension eventually yields the theorem of the weak tropical Nullstellensatz and gives an algebraic interpretation of the tropical Nullstellensatz.

2 0 M ay 1 99 8 EFFECTIVE NULLSTELLENSATZ FOR ARBITRARY IDEALS

A general framework for geometric dualities for varieties of algebras

We set up a framework that subsumes many important dualities in mathematics (Birkhoff, Stone, Priestly, Baker-Beynon, etc.) as well as the classical correspondence between polynomial ideals and affine varieties in algebraic geometry. Our main theorems provide a generalisation of Hilbert’s Nullstellensatz to any (possibly infinitary) variety of algebras. The common core of the above dualities be...

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...