The tetrahedron equation and algebraic geometry

Decoding Algebraic Geometry codes by a key equation

A new effective decoding algorithm is presented for arbitrary algebraicgeometric codes on the basis of solving a generalized key equation with the majority coset scheme of Duursma. It is an improvement of Ehrhard’s algorithm, since the method corrects up to the half of the Goppa distance with complexity order O(n), and with no further assumption on the degree of the divisor G.

Algebraic Geometry the Fundamental Ideas of Abstract Algebraic Geometry

ALGEBRAIC GEOMETRY 79 It may be advisable to give a special name to those varieties which admit every (allowable) ground field as field of definition. Obviously, these are the varieties which are defined over the prime field of the given characteristic p. I propose to call them universal varieties. The projective space and the Grassmannian varieties are examples of universal varieties. Another ...

Introductory Geometry: Algebraic Geometry

Zamolodchikov’s Tetrahedron Equation and Hidden Structure of Quantum Groups

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubi...

Poincaré and Algebraic Geometry

A few of the contributions of Poincaré to algebraic geometry are described, with emphasis on his late work on normal functions. A very brief description of some recent works is also given. Although the subject of algebraic geometry was not one of Poincaré's major preoccupations, his work in the field showed characteristic insight and brilliance and certainly has had a lasting effect. I shall br...