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Akihiro Munemasa connects finite geometry, quadratic forms and algebraic combinatorics. His author-written notes The geometry of orthogonal groups over finite fields by Akihiro Munemasa develop geometric objects and associated graphs from quadratic forms over finite fields. The preface makes the purpose explicit: the geometry supplies nontrivial examples for studying graph symmetries.^[ref]

For a concrete comparison over ${\mathbb{F}}_3$, consider $Q_1(x,y)=xy$ and $Q_2(x,y)=x^2+y^2$. The first vanishes on both coordinate axes, giving nonzero isotropic vectors. The second vanishes only at $(0,0)$: a nonzero square is $1$, and $1+1=2\ne 0$ in ${\mathbb{F}}_3$. Thus equal-dimensional nondegenerate quadratic forms can have different Witt indices, here one and zero. The existence or absence of totally isotropic subspaces changes the incidence geometry built from the form. Munemasa's notes turn these algebraic distinctions into explicit geometric and combinatorial constructions.^[ref]

A quadratic form $q$ over a field $k$ is an anisotropic quadratic form if $q(v)=0$ forces $v=0$. An isotropic quadratic form instead has a nonzero zero vector. This is a property of the form together with its coefficient field, rather than just the displayed polynomial.^[ref]

For example, $q(x,y)=x^2-2y^2$ is anisotropic over $\mathbb{Q}$. If a nonzero rational solution had $y=0$, then $x=0$ too; otherwise $x/y$ would be a rational square root of two. Over $\mathbb{R}$, however, $(\sqrt{2},1)$ is a nonzero zero vector. The coordinate change $u=x+\sqrt{2}y$, $v=x-\sqrt{2}y$ gives $q=uv$, a hyperbolic plane of a quadratic form.

This example also separates anisotropy from positive definiteness: the rational form takes the positive value $1$ and the negative value $-2$, despite having no nonzero rational zero. In a Witt decomposition, an anisotropic remainder is what remains after all hyperbolic planes have been split off. Its Witt index is zero. In characteristic two, one must test vanishing of the quadratic form itself; properties of the associated polar bilinear form alone do not supply the same test.

Over a finite field ${\mathbb{F}}_q$, an elliptic quadric $Q^{-}(2n+1,q)$ is the zero set in projective space $\mathrm{PG}(2n+1,q)$ of a nonsingular quadratic form on a $(2n+2)$-dimensional vector space whose Witt index is $n$. Thus the largest vector subspaces on which the form vanishes identically have dimension $n$, one less than for a hyperbolic form in the same dimension. The word elliptic describes this type of quadratic form.

The especially important case $Q^{-}(3,q)$ contains $q^2+1$ points and no projective line. A line therefore meets it in at most two points, making it an example of an ovoid. These properties make elliptic quadrics useful configurations in finite geometry and in the study of incidence ranks.^[ref]

For example, in $\mathrm{PG}(3,3)$ the equation $x_0^2+x_1^2+x_2^2=x_3^2$ gives an elliptic quadric. With $x_3=1$ there are six solutions, since exactly one of the first three coordinates is nonzero. With $x_3=0$ all three must be nonzero; the eight vectors give four projective points after identifying scalar multiples. There are ten points altogether, as $3^2+1$ predicts.

For a finite-dimensional nonsingular quadratic form $q$ over a field of characteristic different from two, a Witt decomposition is an orthogonal splittingwhere $H(u,v)=uv$ is a hyperbolic plane of a quadratic form and $q_{\mathrm{an}}$ is an anisotropic quadratic form. The integer $r$ is the Witt index. Both $r$ and the isometry class of the anisotropic part are determined by $q$, although the particular subspaces realizing the splitting need not be unique.^[ref]

Over $\mathbb{Q}$, take $q(x,y,z)=x^2+y^2-z^2$. Set $x=(u+v)/2$, $z=(u-v)/2$ and $y=w$. Thenso $q\cong H\perp ⟨1⟩$. The one-dimensional remainder is anisotropic and the Witt index is one. This is an isometry of quadratic spaces, stronger than merely expressing the underlying vector space as a direct sum.

The construction starts from an isotropic vector, finds a partner with nonzero polar pairing, and splits off the hyperbolic plane they generate. Repeating the process ends when no nonzero isotropic vector remains. The decomposition makes the isotropic directions and the residual arithmetic obstruction visible separately.

The Witt index of a nondegenerate quadratic form $q$ on a finite-dimensional vector space $V$ is the largest dimension of a totally isotropic subspace: a linear subspace $U$ for which $q(u)=0$ for every $u\in U$. In characteristic different from two, the Witt decomposition writeswhere $H$ is a hyperbolic plane of a quadratic form, $q_{\mathrm{an}}$ is an anisotropic quadratic form, and $r$ is the Witt index. The anisotropic part has no nonzero vector on which the form vanishes.^[ref]

For the real form $x_1^2+x_2^2-x_3^2-x_4^2$, the subspace spanned by $(1,0,1,0)$ and $(0,1,0,1)$ is totally isotropic. Projection onto the two negative coordinates is injective on any totally isotropic subspace, giving the matching upper bound two. More generally, a real form of signature $(a,b)$ has Witt index $\min (a,b)$.

The base field matters: $x^2+y^2$ has index zero over $\mathbb{R}$ but index one over $\mathbb{C}$, using the line spanned by $(1,i)$. In characteristic two, vanishing of the associated polar form on $U$ alone is insufficient; the condition remains $q{|}_U=0$. Over a finite field, the two nondegenerate types in dimension $2m$ have Witt indices $m$ and $m-1$.^[ref]