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# 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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