6. Gröbner bases and Standard bases

Abstract

• 6.1 Introduction

  • 6.1.1 Algorithms and real numbers

• 6.2 Motivation: Gröbner bases

  • 6.2.1 Term orders for Gröbner bases

  • 6.2.2 Basic question

  • 6.2.3 Rephrasing the basic question

  • 6.2.4 A criterion for Gröbner bases

• 6.3 Standard bases

  • 6.3.1 Overview

  • 6.3.2 Definitions

  • 6.3.3 Setup

  • 6.3.4 Normal form property

  • 6.3.5 The standard map theorem

  • 6.3.6 Normal form algorithm

  • 6.3.7 Reduced normal forms

• 6.4 Instances of standard bases

  • 6.4.1 Gröbner bases

  • 6.4.2 Standard bases for submodules

  • 6.4.3 Standard bases for subalgebras

  • 6.4.4 Standard bases for modules over subalgebras

  • 6.4.5 Left-Right tangent space

• 6.5 The ring of formal power series

  • 6.5.1 Three approaches

  • 6.5.2 Existence of a normal form for formal power series

  • 6.5.3 Truncated formal power series