  
  [1X3 [33X[0;0YBackground Theory on Forms[133X[101X
  
  [33X[0;0YIn  this chapter we give a very brief overview of the theory of sesquilinear
  and quadratic forms. The reader can find more in the texts: Cameron [Cam00],
  Taylor [Tay92], Aschbacher [Asc00], or Kleidman and Liebeck [KL90].[133X
  
  
  [1X3.1 [33X[0;0YSesquilinear forms[133X[101X
  
  [33X[0;0YA  [13Xsesquilinear form[113X on an [22Xn[122X-dimensional vector space [22XV[122X over a field [22XF[122X, is a
  map [22Xf[122X from [22XV× V[122X to [22XF[122X which is linear in the first coordinate, but semilinear
  in  the  second  coordinate;  that  is, there is a field automorphism [22Xα[122X (the
  [13Xcompanion automorphism[113X of [22Xf[122X) such that[133X
  
  
  [24X[33X[0;6Yf(v,\lambda w)=\lambda^\alpha f(v,w),[133X
  
  [124X
  
  [33X[0;0Yfor all [22Xv,w ∈ V[122X and [22Xλ ∈ F[122X. If [22Xα[122X is the identity, then [22Xf[122X is [13Xbilinear[113X.[133X
  
  [33X[0;0YA  bilinear  form  is  [13Xreflexive[113X  if  [22Xf(v,w)=0 ⇒ f(w,v)=0[122X for all [22Xv,w ∈ V[122X. A
  bilinear  form  is  [13Xsymmetric[113X  if [22Xf(v,w)=f(w,v)[122X for all [22Xv,w ∈ V[122X. It is clear
  that  a symmetric bilinear form is reflexive. A bilinear form is [13Xalternating[113X
  if [22Xf(v,v)=0[122X for all [22Xv ∈ V[122X. Using the linearity to compute [22Xf(v+w,v+w)[122X, we see
  that  an  alternating form is also reflexive. When the characteristic of the
  field  differs  from  2,  an alternating form [22Xf[122X can also be characterised as
  [22Xf(v,w)  =  -f(w,v)[122X  for  all  [22Xv,w  ∈  V[122X.  It can be proved (see Chapter 7 of
  [Tay92])  that  symmetric  and  alternating  bilinear  forms  are  the  only
  reflexive bilinear forms.[133X
  
  [33X[0;0YFor a given sesquilinear form [22Xf[122X, the choice of the basis determines uniquely
  an [22Xn× n[122X matrix [22XM[122X such that [22Xf(v, w) = v M w^α}^T.[122X[133X
  
  [33X[0;0YThis  matrix  is also called the [13XGram matrix[113X of [22Xf[122X. Given a sesquilinear form
  [22Xf[122X,  we  will denote its Gram matrix by [22XM_f.[122X In [5XForms[105X, sesquilinear forms can
  be  constructed  using matrices or polynomials, where we always suppose that
  the  basis  of the vector space is the standard basis (i.e., the rows of the
  identity matrix).[133X
  
  [33X[0;0YOne  proves easily that a bilinear form [22Xf[122X is symmetric if and only if [22XM_f[122X is
  a  symmetric  matrix,  i.e.,  [22XM_f^T=M_f,[122X  and  that  a  bilinear  form  [22Xf[122X is
  alternating  if and only if [22XM_f[122X is a skew symmetric matrix, i.e., [22XM_f^T=-M_f[122X
  .  When  the characteristic of the field is two, the condition that [22Xf(v,v)=0[122X
  for all [22Xv ∈ V[122X implies [22XM_f^T=M_f[122X [12Xand[112X [22X(M_ii)=0[122X for all [22Xi[122X (and where the matrix
  [22XM_f = (M_ij)[122X ). Since any skew symmetric odd dimensional matrix is singular,
  it  follows  that  an alternating form of an odd dimensional vector space is
  degenerate.[133X
  
  [33X[0;0YA  sesquilinear  form [22Xf[122X is [13Xhermitian[113X (n.b., [13Xconjugate-symmetric[113X in [CCN+85])
  if  [22Xf(v,w)=f(w,v)^α[122X  holds  for  all vectors [22Xv,w[122X, with [22Xα[122X an involutory field
  automorphism  only  dependent  on  [22Xf[122X.  Again, it can be easily proved that a
  sesquilinear  form  [22Xf[122X  is  hermitian  if  and only if [22XM_f^T = M_f^α[122X (i.e., a
  hermitian  matrix).  It  is proved (see Chapter 7 of [Tay92]) that hermitian
  forms  are  the  only  reflexive  sesquilinear  forms that are not bilinear.
  Hence,  in  general,  all  reflexive  sesquilinear forms are known, they are
  either  hermitian  or  bilinear,  and  in  the  latter case, they are either
  symmetric or alternating (again, see Chapter 7 of [Tay92]).[133X
  
  [33X[0;0YIn  [5XForms[105X, only the construction of [12Xreflexive[112X sesquilinear forms is allowed.
  An   error  message  will  be  displayed  if  any  attempt  to  construct  a
  non-reflexive  sesquilinear  form is made. As a consequence, the Gram Matrix
  of  a  sesquilinear  form  is  always  a  symmetric,  a  skew symmetric or a
  hermitian  matrix.  From  now on, the notion of a ``sesquilinear form'' will
  always refer to a ``reflexive sesquilinear form''.[133X
  
  [33X[0;0YGiven a sesquilinear form [22Xf[122X, two vectors [22Xv[122X and [22Xw[122X are [13Xorthogonal[113X with respect
  to  [22Xf[122X  if  [22Xf(v,w) = 0[122X. Note that the reflexivity makes orthogonality between
  two  vectors  a  symmetric  relation.  A  vector  [22Xv[122X  is  called [13Xisotropic[113X if
  [22Xf(v,v)=0[122X.  The  [13Xradical[113X  of  [22Xf[122X  (n.b.,  [13Xkernel[113X  in [CCN+85]) is the subspace
  consisting of vectors which are orthogonal to every vector. That is,[133X
  
  
  [24X[33X[0;6YRad(f) = \{v \in V | f(v,w) = 0,\, \forall w \in V\},[133X
  
  [124X
  
  [33X[0;0Yand  we  say  that  [22Xf[122X  is  [13Xnon-degenerate[113X  if  its  radical  is trivial (and
  [13Xdegenerate[113X otherwise).[133X
  
  [33X[0;0YGiven  a  subspace  [22XW[122X, we denote the set of vectors of [22XV[122X orthogonal with all
  vectors of [22XW[122X by [22XW^perp[122X . We call a subspace [22XW[122X [13Xtotally isotropic[113X with respect
  to [22Xf[122X if [22XW[122X is contained in [22XW^perp[122X , i.e.[133X
  
  
  [24X[33X[0;6Yf(v,w) = 0,\, \forall v,w \in W.[133X
  
  [124X
  
  [33X[0;0YSuppose that [22Xf[122X is a non-degenerate sesquilinear form. The [13XWitt index[113X of [22Xf[122X is
  the maximum dimension of a totally isotropic subspace with respect to [22Xf[122X.[133X
  
  [33X[0;0YLet  [22Xf[122X  be  a  sesquilinear  form on [22XV(n,q)[122X, with radical [22XR[122X, a [22Xk[122X-dimensional
  subspace  of  [22XV(n,q)[122X,  [22X0  ≤ k ≤ n[122X. Then [22Xf[122X induces a non-degenerate form [22Xg[122X on
  [22XV/R[122X.  When  [22Xdim(R)=0[122X,  then  [22Xg=f[122X  and  [22Xf[122X  is non-degenerate. Notice that all
  totally  isotropic  subspaces  of  maximal  dimension of a degenerate form [22Xf[122X
  contain  the radical of [22Xf[122X. In [5XForms[105X, the notion Witt index will [12Xalways refer
  to  the  induced  non-degenerate  form[112X  [22Xg[122X. Hence, given a degenerate form [22Xf[122X,
  computing  its  Witt index will return the Witt index of the induced form [22Xg[122X.
  [12XThis  also  holds  for  the notions elliptic, parabolic and hyperbolic for a
  bilinear form, which are notions defined using the Witt index, see below[112X.[133X
  
  [33X[0;0YWe  end  this  section  with  a short description of the conventions used in
  [5XForms[105X  for  the notions orthogonal, symplectic, pseudo, hyperbolic, elliptic
  and  parabolic. We call a form [22Xf[122X [13Xsymplectic[113X if and only if [22Xf[122X is alternating.
  When  the characteristic of the field is odd, we call a form [22Xf[122X [13Xorthogonal[113X if
  and  only  [22Xf[122X is symmetric, and when the characteristic of the field is even,
  we  call  a form [22Xf[122X [13Xpseudo[113X if and only if [22Xf[122X is symmetric but not alternating.
  This  terminology  is related to the theory of polar spaces, and in the case
  of  orthogonal  forms, we adopt the terms [13Xhyperbolic[113X, [13Xelliptic[113X and [13Xparabolic[113X
  for  the  three different isometry types of orthogonal forms. From the point
  of  view  of  matrix groups, these three types correspond as follows. Recall
  that,  as  explained  above,  the Witt index refers to the Witt index of the
  [12Xinduced non-degenerate form[112X [22Xg[122X when [22Xf[122X is degenerate.[133X
  
        ───────────┬──────────────────────┬──────────────────────────────────────────────────────────  
        Hyperbolic │ Orthogonal of + type │ [22XV/Rad(f)[122X has even dimension, [22Xg[122X has maximal Witt index       
        Elliptic   │ Orthogonal of - type │ [22XV/Rad(f)[122X has even dimension, [22Xg[122X has non-maximal Witt index   
        Parabolic  │ Orthogonal of o type │ [22XV/Rad(f)[122X has odd dimension                                  
        ───────────┴──────────────────────┴──────────────────────────────────────────────────────────  
  
       [1XTable:[101X Posibilites for an orthogonal form [22Xf[122X on a vector space [22XV[122X
  
  
  
  [1X3.1-1 [33X[0;0YExamples[133X[101X
  
  [33X[0;0YThe  examples we present in this section do not demonstrate the entire suite
  of  operations  entailed  in  [5XForms[105X.  They are intended to allow the user to
  become   familiar   with   particular  aspects  of  this  package.  All  the
  functionality  for  sesquilinear  forms will be listed in detail in the next
  chapter.[133X
  
  [33X[0;0YWe try to construct a bilinear form...[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat := [[1,0,0],[0,1,4],[1,2,1]]*Z(5)^0;[127X[104X
    [4X[28X[ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, Z(5)^2 ], [128X[104X
    [4X[28X  [ Z(5)^0, Z(5), Z(5)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xform := BilinearFormByMatrix(mat,GF(5));[127X[104X
    [4X[28XError, Invalid Gram matrix[128X[104X
    [4X[28X called from[128X[104X
    [4X[28XBilinearFormByMatrixOp( MutableCopyMat( m ), f [128X[104X
    [4X[28X ) at ./pkg/forms/lib/forms.gi:164 called from[128X[104X
    [4X[28X<function "unknown">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 8 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit; [127X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is clear that the matrix used is not defining a reflexive bilinear form,
  which causes the system to generate the error message.[133X
  
  [33X[0;0YWe  construct now a reflexive bilinear form. We investigate also the radical
  of the form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat := [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,-1]]*Z(9)^0;[127X[104X
    [4X[28X[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [128X[104X
    [4X[28X  [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3) ] ][128X[104X
    [4X[25Xgap>[125X [27Xform := BilinearFormByMatrix(mat,GF(9));[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(form);[127X[104X
    [4X[28XBilinear form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28X 1 . . .[128X[104X
    [4X[28X . 1 . .[128X[104X
    [4X[28X . . 1 .[128X[104X
    [4X[28X . . . 2[128X[104X
    [4X[25Xgap>[125X [27XIsReflexiveForm(form);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricForm(form);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAlternatingForm(form);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xr := RadicalOfForm(form);[127X[104X
    [4X[28X<vector space over GF(3^2), with 0 generators>[128X[104X
    [4X[25Xgap>[125X [27XDimension(r);[127X[104X
    [4X[28X0 [128X[104X
  [4X[32X[104X
  
  [33X[0;0YDegenerate  forms  are  allowed.  As  an example we construct an alternating
  bilinear form on an odd dimensional vector space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat := [[0,0,-2],[0,0,1],[2,-1,0]]*Z(7)^0;[127X[104X
    [4X[28X[ [ 0*Z(7), 0*Z(7), Z(7)^5 ], [ 0*Z(7), 0*Z(7), Z(7)^0 ], [128X[104X
    [4X[28X  [ Z(7)^2, Z(7)^3, 0*Z(7) ] ][128X[104X
    [4X[25Xgap>[125X [27Xform := BilinearFormByMatrix(mat,GF(7));[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(form);[127X[104X
    [4X[28XBilinear form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28X . . 5[128X[104X
    [4X[28X . . 1[128X[104X
    [4X[28X 2 6 .[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricForm(form);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsAlternatingForm(form);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xr := RadicalOfForm(form);[127X[104X
    [4X[28X<vector space over GF(7), with 1 generators>[128X[104X
    [4X[25Xgap>[125X [27XDimension(r);[127X[104X
    [4X[28X1 [128X[104X
  [4X[32X[104X
  
  [33X[0;0YWhen  the characteristic of the field equals two, alternating forms are also
  symmetric. We construct an example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat := [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[127X[104X
    [4X[25X>[125X [27X        [0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]*Z(16)^0;[127X[104X
    [4X[28X[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ] ][128X[104X
    [4X[25Xgap>[125X [27Xform := BilinearFormByMatrix(mat,GF(16));[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(form);[127X[104X
    [4X[28XBilinear form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28X . 1 . . . .[128X[104X
    [4X[28X 1 . . . . .[128X[104X
    [4X[28X . . . . . 1[128X[104X
    [4X[28X . . . . 1 .[128X[104X
    [4X[28X . . . 1 . .[128X[104X
    [4X[28X . . 1 . . .[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricForm(form);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAlternatingForm(form);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDegenerateForm(form);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XWittIndex(form);[127X[104X
    [4X[28X3 [128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo define a hermitian form, we need a matrix and the companion automorphism.
  Since  this  automorphism has order 2, it exists and is unique if the ground
  field  has  square order. In the next example, the chosen matrix is somewhat
  special. Together with the companion automorphism, it determines a hermitian
  sesquilinear  form.  Without  the  companion  automorphism, it determines an
  alternating bilinear form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat := [[0*Z(5),0*Z(5),0*Z(25),Z(25)^3],[0*Z(5),0*Z(5),Z(25)^3,0*Z(25)],[127X[104X
    [4X[25X>[125X [27X        [0*Z(5),-Z(25)^3,0*Z(5),0*Z(5)],[-Z(25)^3,0*Z(5),0*Z(25),0*Z(25)]];[127X[104X
    [4X[28X[ [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5^2)^3 ], [ 0*Z(5), 0*Z(5), Z(5^2)^3, 0*Z(5) ], [128X[104X
    [4X[28X  [ 0*Z(5), Z(5^2)^15, 0*Z(5), 0*Z(5) ], [128X[104X
    [4X[28X  [ Z(5^2)^15, 0*Z(5), 0*Z(5), 0*Z(5) ] ][128X[104X
    [4X[25Xgap>[125X [27Xform := HermitianFormByMatrix(mat,GF(25));[127X[104X
    [4X[28X< hermitian form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(form);[127X[104X
    [4X[28XHermitian form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28Xz = Z(25)[128X[104X
    [4X[28X    .    .    .  z^3[128X[104X
    [4X[28X    .    .  z^3    .[128X[104X
    [4X[28X    . z^15    .    .[128X[104X
    [4X[28X z^15    .    .    .[128X[104X
    [4X[25Xgap>[125X [27XWittIndex(form);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xform2 := BilinearFormByMatrix(mat,GF(25));[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(form2);[127X[104X
    [4X[28XBilinear form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28Xz = Z(25)[128X[104X
    [4X[28X    .    .    .  z^3[128X[104X
    [4X[28X    .    .  z^3    .[128X[104X
    [4X[28X    . z^15    .    .[128X[104X
    [4X[28X z^15    .    .    .[128X[104X
    [4X[25Xgap>[125X [27XIsAlternatingForm(form2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDisplay(IsometricCanonicalForm(form));[127X[104X
    [4X[28XHermitian form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28X 1 . . .[128X[104X
    [4X[28X . 1 . .[128X[104X
    [4X[28X . . 1 .[128X[104X
    [4X[28X . . . 1[128X[104X
    [4X[28XWitt Index: 2[128X[104X
    [4X[25Xgap>[125X [27XDisplay(IsometricCanonicalForm(form2));[127X[104X
    [4X[28XBilinear form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28X . 1 . .[128X[104X
    [4X[28X 4 . . .[128X[104X
    [4X[28X . . . 1[128X[104X
    [4X[28X . . 4 .[128X[104X
    [4X[28XWitt Index: 2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  continue the previous example by exploring a little bit the sesquilinear
  form  [3Xform[103X,  and  hence  demonstrate  some of the functionality of the [5XForms[105X
  package.  Eventually,  we  find  a 2-dimensional totally isotropic subspace,
  which  lets  us conclude that the Witt index of [22Xform[122X is at least 2, which is
  confirmed afterwards by calling the appropriate function.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV := GF(25)^4;[127X[104X
    [4X[28X( GF(5^2)^4 )[128X[104X
    [4X[25Xgap>[125X [27Xu := [Z(5)^0,Z(5^2)^11,Z(5)^3,Z(5^2)^13 ];[127X[104X
    [4X[28X[ Z(5)^0, Z(5^2)^11, Z(5)^3, Z(5^2)^13 ][128X[104X
    [4X[25Xgap>[125X [27X[u,u]^form;[127X[104X
    [4X[28X0*Z(5)[128X[104X
    [4X[25Xgap>[125X [27Xv := [Z(5)^0,Z(5^2)^5,Z(5^2),Z(5^2)^13 ];[127X[104X
    [4X[28X[ Z(5)^0, Z(5^2)^5, Z(5^2), Z(5^2)^13 ][128X[104X
    [4X[25Xgap>[125X [27X[v,v]^form;                                     [127X[104X
    [4X[28X0*Z(5)[128X[104X
    [4X[25Xgap>[125X [27X[u,v]^form;[127X[104X
    [4X[28XZ(5^2)^7[128X[104X
    [4X[25Xgap>[125X [27X([v,u]^form)^5;[127X[104X
    [4X[28XZ(5^2)^7[128X[104X
    [4X[25Xgap>[125X [27Xw := [Z(5^2)^21,Z(5^2)^19,Z(5^2)^4,Z(5)^3 ];[127X[104X
    [4X[28X[ Z(5^2)^21, Z(5^2)^19, Z(5^2)^4, Z(5)^3 ][128X[104X
    [4X[25Xgap>[125X [27X[w,w]^form;[127X[104X
    [4X[28XZ(5)[128X[104X
    [4X[25Xgap>[125X [27Xv := [Z(5)^0,Z(5^2)^10,Z(5^2)^15,Z(5^2)^3 ];[127X[104X
    [4X[28X[ Z(5)^0, Z(5^2)^10, Z(5^2)^15, Z(5^2)^3 ][128X[104X
    [4X[25Xgap>[125X [27Xu := [Z(5)^3,Z(5^2)^9,Z(5^2)^4,Z(5^2)^16 ];[127X[104X
    [4X[28X[ Z(5)^3, Z(5^2)^9, Z(5^2)^4, Z(5^2)^16 ][128X[104X
    [4X[25Xgap>[125X [27Xw := [Z(5)^2,Z(5^2)^9,Z(5^2)^23,Z(5^2)^11 ];[127X[104X
    [4X[28X[ Z(5)^2, Z(5^2)^9, Z(5^2)^23, Z(5^2)^11 ][128X[104X
    [4X[25Xgap>[125X [27X[u,v]^form;[127X[104X
    [4X[28X0*Z(5)[128X[104X
    [4X[25Xgap>[125X [27X[u,w]^form;[127X[104X
    [4X[28X0*Z(5)[128X[104X
    [4X[25Xgap>[125X [27X[v,w]^form;[127X[104X
    [4X[28X0*Z(5)[128X[104X
    [4X[25Xgap>[125X [27Xs := Subspace(V,[v,u,w]);[127X[104X
    [4X[28X<vector space over GF(5^2), with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XDimension(s);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XWittIndex(form);[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YQuadratic forms[133X[101X
  
  [33X[0;0YA [13Xquadratic form[113X on an [22Xn[122X-dimensional vector space [22XV[122X over a field [22XF[122X, is a map
  [22XQ[122X from [22XV[122X to [22XF[122X satisfying the following two conditions:[133X
  
  
  [24X[33X[0;6YQ(\lambda v) = \lambda^2 Q(v),\, \forall \lambda \in F, \forall v \in V,[133X
  
  [124X
  
  [33X[0;0Yand, the map [22Xf[122X defined from [22XV× V[122X to [22XF[122X as follows,[133X
  
  
  [24X[33X[0;6Yf(v,w) := Q(v+w) - Q(v) - Q(w),[133X
  
  [124X
  
  [33X[0;0Yis a bilinear form on [22XV[122X. From this definition it follows that [22Xf(v,v) = Q(2v)
  - 2Q(v) = 2Q(v)[122X.[133X
  
  [33X[0;0YThe  associated  bilinear  form  [22Xf[122X  (which  is called the [13Xpolar form[113X of [22XQ[122X in
  [CCN+85]) is clearly reflexive. When the characteristic of the field is odd,
  it is clear that [22Xf[122X is a symmetric bilinear form. The equation [22Xf(v,v) = 2Q(v)[122X
  allows  us  to  reconstruct  the  quadratic form from the bilinear form, and
  hence  there  is  a  one-to-one  correspondence  between quadratic forms and
  symmetric bilinear forms. When the characteristic of the field equals 2, the
  bilinear  form  [22Xf[122X  is  alternating  (from the fact that [22Xf(v,v) = 2Q(v) = 0[122X).
  Note,  however,  that  different  quadratic  forms  can  determine  the same
  alternating form.[133X
  
  [33X[0;0YAs  in  the  case  of  sesquilinear  forms,  we will associate a matrix to a
  quadratic  form. Choosing a basis of the vector space [22XV[122X, it is clear that an
  [22Xn  ×  n[122X  matrix determines the quadratic form completely. In [5XForms[105X, the [13XGram
  matrix[113X of a quadratic form is always an upper triangle matrix [22XM[122X, such that[133X
  
  
  [24X[33X[0;6YQ(v) = vMv^T,[133X
  
  [124X
  
  [33X[0;0Ywhere  the basis of [22XV[122X is the standard basis. Although the Gram matrix stored
  with  the  quadratic  form  is  always an upper triangle matrix, the user is
  allowed  to  use any matrix to define the quadratic form, since any matrix [22XM[122X
  defines  a  quadratic  form  [22XQ(v)  :=  vMv^T[122X  .  During the construction, an
  appropriate upper triangle matrix is computed and stored as the Gram matrix.
  So the Gram matrix of the associated bilinear form is [22XM+M^T[122X.[133X
  
  [33X[0;0YThe   associated   bilinear  form  could  be  used  to  define  the  notions
  ``isotropic'',  ``totally isotropic'' and ``non-degenerate'', however, under
  these restrictions the geometry of quadratic forms in even characteristic is
  lost.  In  most  of  the  literature,  these  notions  refer  indeed  to the
  associated  bilinear  form,  and  the  notion of ``singularity'' is added to
  regain the geometrical structure.[133X
  
  [33X[0;0YIn  [5XForms[105X,  we use the above described approach. This means that a vector is
  isotropic  if  and  only  if  it is isotropic with respect to the associated
  bilinear  form. A subspace is totally isotropic if and only if it is totally
  isotropic  with  respect  to  the  associated bilinear form, and we call the
  quadratic  form  degenerate  if  and only if the associated bilinear form is
  degenerate.[133X
  
  [33X[0;0YA  vector  [22Xv[122X is called [13Xsingular[113X with relation to the quadratic form [22XQ[122X if and
  only  if [22XQ(v)=0[122X. two vectors [22Xv[122X and [22Xw[122X are [13Xorthogonal[113X with respect to [22XQ[122X if and
  only  if they are orthogonal with respect to the associated bilinear form [22Xf[122X.
  The  [13Xradical[113X  of the quadratic form [22XQ[122X, is the intersection of the set of all
  singular  vectors  with  relation  to  [22XQ[122X  and  the radical of the associated
  bilinear form [22Xf[122X, i.e.[133X
  
  
  [24X[33X[0;6YRad(Q) = \{v \in V | Q(v) = 0\,\, \mathrm{and}\,\, v \in Rad(f)\}.[133X
  
  [124X
  
  [33X[0;0YWe  call a quadratic form [22XQ[122X [13Xnon-singular[113X if and only if the radical contains
  only the zero vector, and [13Xsingular[113X otherwise.[133X
  
  [33X[0;0YA  subspace  [22XW[122X of the vector space is called [13Xtotally singular[113X if and only if
  all vectors of [22XW[122X are singular, i.e., [22XQ[122X vanishes totally on [22XW[122X. Necessarily, a
  totally  singular  subspace  is  also totally isotropic with relation to the
  associated  bilinear  form  [22Xf[122X,  but  the  converse  is  only  true  when the
  characteristic of the field is odd.[133X
  
  [33X[0;0YSuppose  now that [22XQ[122X is a non-singular quadratic form. The [13XWitt index[113X of [22XQ[122X is
  the maximum dimension of a totally singular subspace with respect to [22XQ[122X.[133X
  
  [33X[0;0YLet  [22XQ[122X  be  a  quadratic  form  on  [22XV(n,q)[122X,  with radical [22XR[122X, a [22Xk[122X-dimensional
  subspace of [22XV(n,q)[122X, [22X0 ≤ k ≤ n[122X. Then [22XQ[122X induces a non-singular form [22XQ'[122X on [22XV/R[122X.
  When  [22Xdim(R)=0[122X,  then  [22XQ=Q'[122X  and  [22XQ[122X is non-singular. Notice that all totally
  singular  subspaces  of  maximal  dimension of a singular form [22XQ[122X contain the
  radical  of  [22XQ[122X.  In  [5XForms[105X,  the  notion Witt index will [12Xalways refer to the
  induced  non-singular form[112X [22XQ'[122X. Hence, given a singular form [22XQ[122X, computing its
  Witt  index  will  return  the  Witt index of the induced form [22XQ'[122X. [12XThis also
  holds  for  the  notions  elliptic, parabolic and hyperbolic for a quadratic
  form, which are notions defined using the Witt index, see below[112X.[133X
  
  [33X[0;0YThe terminology[13Xhyperbolic[113X, [13Xelliptic[113X and [13Xparabolic[113X is also used for quadratic
  forms,  and  is defined analogously as for the bilinear forms using the Witt
  index.  Also  in the case of quadratic forms, this terminology is related to
  the  theory of polar spaces. Recall that, as explained above, the Witt index
  refers  to  the  Witt  index  of  the [12Xinduced non-singular form[112X [22XQ[122X when [22XQ'[122X is
  singular.[133X
  
        ───────────┬──────────────────────┬───────────────────────────────────────────────────────────  
        Hyperbolic │ Orthogonal of + type │ [22XV/Rad(Q)[122X has even dimension, [22XQ'[122X has maximal Witt index       
        Elliptic   │ Orthogonal of - type │ [22XV/Rad(Q)[122X has even dimension, [22XQ'[122X has non-maximal Witt index   
        Parabolic  │ Orthogonal of o type │ [22XV/Rad(Q)[122X has odd dimension                                   
        ───────────┴──────────────────────┴───────────────────────────────────────────────────────────  
  
       [1XTable:[101X Posibilites for a quadratic form [22XQ[122X on a vector space [22XV[122X
  
  
  [33X[0;0YFrom  the above definitions, it follows that, when the characteristic of the
  field  differs from 2, a quadratic form [22XQ[122X is non-singular if and only if its
  associated bilinear form [22Xf[122X is non-degenerate. When the characteristic of the
  field  is  2,  one can easily construct non-singular quadratic forms, with a
  degenerate  associated  bilinear  form.  We  will  give  an  example of this
  situation in the next section.[133X
  
  
  [1X3.2-1 [33X[0;0YExamples[133X[101X
  
  [33X[0;0YWe construct some quadratic forms to demonstrate some funcionality of [5XForms[105X.
  As  in  the previous example section, they are intended to allow the user to
  gain  some  familiarity.  All  the functionality for quadratic forms will be
  listed in detail in the next chapter.[133X
  
  [33X[0;0YThe user can construct quadratic forms using any matrix (provided it has the
  right  dimension).  The  Gram  matrix  is always stored as an upper triangle
  matrix, as explained above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV := GF(4)^3;                           [127X[104X
    [4X[28X( GF(2^2)^3 )[128X[104X
    [4X[25Xgap>[125X [27Xmat := [[Z(2^2)^2,Z(2^2),Z(2^2)^2],[Z(2^2)^2,Z(2)^0,Z(2)^0],[127X[104X
    [4X[25X>[125X [27X        [0*Z(2),Z(2)^0,0*Z(2)]];[127X[104X
    [4X[28X[ [ Z(2^2)^2, Z(2^2), Z(2^2)^2 ], [ Z(2^2)^2, Z(2)^0, Z(2)^0 ], [128X[104X
    [4X[28X  [ 0*Z(2), Z(2)^0, 0*Z(2) ] ][128X[104X
    [4X[25Xgap>[125X [27Xqform := QuadraticFormByMatrix(mat, GF(4));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay( qform );[127X[104X
    [4X[28XQuadratic form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28Xz = Z(4)[128X[104X
    [4X[28X z^2   1 z^2[128X[104X
    [4X[28X   .   1   .[128X[104X
    [4X[28X   .   .   .[128X[104X
    [4X[25Xgap>[125X [27XPolynomialOfForm( qform );[127X[104X
    [4X[28XZ(2^2)^2*x_1^2+x_1*x_2+Z(2^2)^2*x_1*x_3+x_2^2 [128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  previous  example,  we  saw  how  we used a polynomial to display a
  quadratic  form.  Conversely, [5XForms[105X allows the user to construct (quadratic)
  forms using a polynomial.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr := PolynomialRing(GF(8),4);[127X[104X
    [4X[28XGF(2^3)[x_1,x_2,x_3,x_4][128X[104X
    [4X[25Xgap>[125X [27Xpoly := r.1*r.2+r.3*r.4;[127X[104X
    [4X[28Xx_1*x_2+x_3*x_4[128X[104X
    [4X[25Xgap>[125X [27Xqform := QuadraticFormByPolynomial(poly, r);[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(qform);[127X[104X
    [4X[28XQuadratic form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28X . 1 . .[128X[104X
    [4X[28X . . . .[128X[104X
    [4X[28X . . . 1[128X[104X
    [4X[28X . . . .[128X[104X
    [4X[28XPolynomial: x_1*x_2+x_3*x_4[128X[104X
    [4X[25Xgap>[125X [27XRadicalOfForm(qform);[127X[104X
    [4X[28X<vector space over GF(2^3), with 0 generators> [128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  construct  now  two  different  quadratic forms with the same associated
  bilinear form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat := [[Z(16)^3,1,0,0],[0,Z(16)^5,0,0],[127X[104X
    [4X[25X>[125X [27X             [0,0,Z(16)^3,1],[0,0,0,Z(16)^12]]*Z(16)^0;[127X[104X
    [4X[28X[ [ Z(2^4)^3, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2^2), 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), Z(2^4)^3, Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^4)^12 ] [128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27Xqform := QuadraticFormByMatrix(mat,GF(16));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay( qform );[127X[104X
    [4X[28XQuadratic form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28Xz = Z(16)[128X[104X
    [4X[28X  z^3    1    .    .[128X[104X
    [4X[28X    .  z^5    .    .[128X[104X
    [4X[28X    .    .  z^3    1[128X[104X
    [4X[28X    .    .    . z^12[128X[104X
    [4X[25Xgap>[125X [27Xmat2 := [[Z(16)^7,1,0,0],[0,0,0,0],[127X[104X
    [4X[25X>[125X [27X             [0,0,Z(16)^2,1],[0,0,0,Z(16)^9]]*Z(16)^0;[127X[104X
    [4X[28X[ [ Z(2^4)^7, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), Z(2^4)^2, Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^4)^9 ] ][128X[104X
    [4X[25Xgap>[125X [27Xqform2 := QuadraticFormByMatrix(mat2, GF(16));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay( qform2 );[127X[104X
    [4X[28XQuadratic form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28Xz = Z(16)[128X[104X
    [4X[28X  z^7    1    .    .[128X[104X
    [4X[28X    .    .    .    .[128X[104X
    [4X[28X    .    .  z^2    1[128X[104X
    [4X[28X    .    .    .  z^9[128X[104X
    [4X[25Xgap>[125X [27Xbiform := AssociatedBilinearForm( qform2 );[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay( biform );[127X[104X
    [4X[28XBilinear form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28X . 1 . .[128X[104X
    [4X[28X 1 . . .[128X[104X
    [4X[28X . . . 1[128X[104X
    [4X[28X . . 1 . [128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  end  with  an example of a non-singular quadratic form with a degenerate
  associated bilinear form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat := [ [ Z(2^2), Z(2^2), Z(2^2), Z(2^2), Z(2^2) ], [127X[104X
    [4X[25X>[125X [27X   [ 0*Z(2), Z(2^2), Z(2^2)^2, 0*Z(2), Z(2)^0 ], [127X[104X
    [4X[25X>[125X [27X   [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], [127X[104X
    [4X[25X>[125X [27X   [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ], [127X[104X
    [4X[25X>[125X [27X   [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xqform := QuadraticFormByMatrix(mat,GF(4));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27XIsSingularForm(qform);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDegenerateForm(qform);[127X[104X
    [4X[28X#I  Testing degeneracy of the *associated bilinear form*[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xbiform := AssociatedBilinearForm(qform);[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(biform);[127X[104X
    [4X[28XBilinear form[128X[104X
    [4X[28XGram Matrix:[128X[104X
    [4X[28Xz = Z(4)[128X[104X
    [4X[28X   . z^1 z^1 z^1 z^1[128X[104X
    [4X[28X z^1   . z^2   .   1[128X[104X
    [4X[28X z^1 z^2   .   1   1[128X[104X
    [4X[28X z^1   .   1   .   1[128X[104X
    [4X[28X z^1   1   1   1   .[128X[104X
    [4X[25Xgap>[125X [27XIsDegenerateForm(biform);[127X[104X
    [4X[28Xtrue [128X[104X
  [4X[32X[104X
  
