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4196 results in General and Classical Physics

Page 78 of 210

  • Preface

  • Wolfram Decker, Technische Universität Kaiserslautern, Germany, Gerhard Pfister, Technische Universität Kaiserslautern, Germany
  • Book:
    A First Course in Computational Algebraic Geometry
    Published online:
    05 March 2013
    Print publication:
    07 February 2013, pp vii-viii

  • Summary

    Preface

    Most of mathematics is concerned at some level with setting up and solving various types of equations. Algebraic geometry is the mathematical discipline which handles solution sets of systems of polynomial equations. These are called algebraic sets.

    By making use of a correspondence which relates algebraic sets to ideals in polynomial rings, problems concerning the geometry of algebraic sets can be translated into algebra. As a consequence, algebraic geometers have developed a multitude of often highly abstract techniques for the qualitative and quantitative study of algebraic sets, without, in the first instance, considering the equations. Modern computer algebra algorithms, on the other hand, allow us to manipulate the equations and, thus, to study explicit examples. In this way, algebraic geometry becomes accessible to experiments. The experimental method, which has proven to be highly successful in number theory, is now also added to the toolbox of the algebraic geometer.

    In these notes, we discuss some of the basic operations in geometry and describe their counterparts in algebra. We explain how the operations can be carried out using computation, and give a number of explicit examples, worked out with the computer algebra system SINGULAR. In this way, our book may serve as a first introduction to SINGULAR, guiding the reader to performing his own experiments.

  • 3 - Sudoku

  • Wolfram Decker, Technische Universität Kaiserslautern, Germany, Gerhard Pfister, Technische Universität Kaiserslautern, Germany
  • Book:
    A First Course in Computational Algebraic Geometry
    Published online:
    05 March 2013
    Print publication:
    07 February 2013, pp 95-100

  • Summary

    In this chapter, we will explain how to solve a Sudoku puzzle using ideas from algebraic geometry and computer algebra. In fact, we will represent the solutions of a Sudoku as the points in the vanishing locus of a polynomial ideal I in 81 variables, and we will show that the unique solution of a well-posed Sudoku can be read off from the reduced Gröbner basis of I. We should point out, however, that attacking a Sudoku can be regarded as a graph coloring problem, with one color for each of the numbers 1, . . . ,9, and that graph theory provides much more efficient methods for solving Sudoko than do Gröbner bases.

    A completed Sudoku is a particular example of what is called a Latin square. A Latin square of order n is an n Ⅹ n square grid whose entries are taken from a set of n different symbols, with each symbol appearing exactly once in each row and each column. For a Sudoku, usually n = 9, and the symbols are the numbers from 1 to 9. In addition to being a Latin square, a completed Sudoku is subject to the condition that each number from 1 to 9 appears exactly once in each of the nine distinguished 3 Ⅹ 3 blocks.

Page 78 of 210