Algebraic geometry over hyperrings

Algebraic Geometry over Lie Algebras

What is algebraic geometry over algebraic systems? Many important relations between elements of a given algebraic system A can be expressed by systems of equations over A. The solution sets of such systems are called algebraic sets over A. Algebraic sets over A form a category, if we take for morphisms polynomial functions in the sense of Definition 6.1 below. As a discipline, algebraic geometr...

Introductory Geometry: Algebraic Geometry

Algebraic Geometry Over Four Rings and the Frontier to Tractability

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set (b) the height of the zero-dimensional part of an algebraic set over C (c) the number of connected components of a semi-algebraic set We also present so...

Algebraic Geometry over Free Groups: Lifting Solutions into Generic Points

In this paper we prove Implicit Function Theorems (IFT) for algebraic varieties defined by regular quadratic equations and, more generally, regular NTQ systems over free groups. In the model theoretic language these results state the existence of very simple Skolem functions for particular ∀∃formulas over free groups. We construct these functions effectively. In noneffective form IFT first appe...

Pure L-Functions from Algebraic Geometry over Finite Fields

This survey gives a concrete and intuitively self-contained introduction to the theory of pure L-functions arising from a family of algebraic varieties defined over a finite field of characteristic p. The standard fundamental questions in any theory of L-functions include the meromorphic continuation, functional equation, Riemann hypothesis (RH for short), order of zeros at special points and t...