GAP 4 Package Forms

Sesquilinear and Quadratic

1.0

May 2007

John Bamberg
e-mail: bamberg@cage.ugent.be
WWW: http://cage.ugent.be/~bamberg
Address:
Department of Pure Mathematics, Ghent University, Galglaan 2, 9000 Ghent, Belgium

Jan De Beule
e-mail: jdebeule@cage.ugent.be
WWW: http://cage.ugent.be/~jdebeule
Address:
Department of Pure Mathematics, Ghent University, Galglaan 2, 9000 Ghent, Belgium

Copyright

(C) 2007 by the authors

This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher.

Contents

1. Introduction
   1.1 Philosophy
   1.2 Overview over this manual
2. Examples
   2.1 A conic of PG(2,8)
   2.2 A form for W(5,3)
3. Background Theory on Forms
   3.1 Sesquilinear forms, dualities, and polarities
      3.1-1 Example
   3.2 Quadratic forms
      3.2-1 Example
   3.3 Morphisms of forms
   3.4 An important convention
      3.4-1 Example
   3.5 Canonical forms
4. Functionality
   4.1 Functions for creating forms
      4.1-1 BilinearFormByMatrix
      4.1-2 QuadraticFormByMatrix
      4.1-3 HermitianFormByMatrix
      4.1-4 BilinearFormByPolynomial
      4.1-5 QuadraticFormByPolynomial
      4.1-6 HermitianFormByPolynomial
   4.2 Attributes and properties of forms
      4.2-1 IsReflexiveForm
      4.2-2 IsAlternatingForm
      4.2-3 IsSymmetricForm
      4.2-4 IsDegenerateForm
      4.2-5 BaseField
      4.2-6 GramMatrix
      4.2-7 WittIndex
      4.2-8 RadicalOfForm
      4.2-9 PolynomialOfForm
      4.2-10 DiscriminantOfForm
   4.3 Functions for changing forms
      4.3-1 BaseChangeToCanonical
      4.3-2 IsometricCanonicalForm
   4.4 Operations on forms
      4.4-1 BaseChangeHomomorphism
      4.4-2 EvaluateForm




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