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Isotropic quadratic form

In mathematics, an isotropic quadratic form is a quadratic form q on a vector space V over a field F (typically of characteristic not equal to 2) that represents zero non-trivially, meaning there exists a non-zero vector v \in V such that q(v) = 0.^[1]^[2] If no such non-zero vector exists, the form is called anisotropic.^[1] Quadratic forms arise in various areas of algebra, geometry, and number theory, where isotropy plays a central role in classification and decomposition theorems.^[2] A key property is that every nonsingular isotropic quadratic form is universal, meaning it represents every element of the base field F.^[2] Moreover, such forms contain a hyperbolic plane as an isometric subspace—a binary form equivalent to \langle 1, -1 \rangle, which is isotropic since it vanishes on vectors like (1,1).^[1]^[3] The study of isotropic quadratic forms is foundational to Witt's theory, which provides a canonical decomposition of any quadratic space into an orthogonal sum of a hyperbolic part (spanned by isotropic subspaces) and an anisotropic kernel.^[2] The Witt index measures the "size" of isotropy, defined as the maximal dimension of a totally isotropic subspace (where q vanishes on every non-zero vector in the subspace).^[2] For binary forms, a nonsingular quadratic form is isotropic if and only if it is equivalent to the hyperbolic plane \langle 1, -1 \rangle.^[1] Isotropy also connects to broader themes, such as the Hasse-Minkowski theorem, which asserts that a quadratic form over the rationals is isotropic if and only if it is isotropic over all local fields (reals and p-adics), facilitating local-global principles in number theory.^[4] In algebraic geometry, isotropic forms relate to the geometry of quadratic hypersurfaces and their singularities, where the isotropic cone \{x \in V \mid q(x) = 0\} defines projective varieties with important arithmetic properties.^[1]

Fundamentals

Definition

In mathematics, a quadratic form q on a finite-dimensional vector space V over a field F of characteristic not equal to 2 is a map q: V \to F that is a homogeneous polynomial of degree 2 in the coordinates of v \in V, or equivalently, there exists a unique symmetric bilinear form B: V \times V \to F such that q(v) = B(v, v) for all v \in V.^[5]^[4] The pair (V, q) is called a quadratic space over F.^[5] A quadratic form q on V is said to be isotropic if there exists a non-zero vector v \in V such that q(v) = 0; such a vector v is called an isotropic vector.^[5]^[4] Equivalently, the set of isotropic vectors, denoted \{ v \in V \mid q(v) = 0, v \neq 0 \}, is non-empty.^[2] In the context of a quadratic space (V, q), the collection of all isotropic vectors forms the non-trivial part of the kernel of q, highlighting the presence of null directions in the form.^[4] The term "isotropic quadratic form" originates from the foundational studies of quadratic forms by Ernst Witt in the 1930s, where emphasis was placed on null vectors in indefinite or non-degenerate forms over arbitrary fields.^[6] A classic example of an isotropic quadratic form arises in the hyperbolic plane over F, where q(x, y) = x y vanishes on the non-zero vector (1, 0).^[5]

Associated Bilinear Forms

To every quadratic form q: V \to K on a vector space V over a field K of characteristic not 2, there is canonically associated a symmetric bilinear form B: V \times V \to K defined by

B(u, v) = \frac{q(u + v) - q(u) - q(v)}{2}.

This association is bijective: given a symmetric bilinear form B, the quadratic form is recovered as q(w) = B(w, w).^[7]^[8] The connection is captured by the polarization identities:

q(u + v) = q(u) + q(v) + 2B(u, v), \quad q(u - v) = q(u) + q(v) - 2B(u, v).

These identities allow the bilinear form B to encode the quadratic structure, facilitating computations such as orthogonality and norms.^[7] A quadratic form q is isotropic if there exists a nonzero vector v \in V such that q(v) = 0; equivalently, B(v, v) = 0 for some v \neq 0, meaning B admits a nontrivial isotropic vector. The radical of B, defined as \mathrm{rad}(B) = \{ w \in V \mid B(w, z) = 0 \ \forall z \in V \}, consists of vectors that are orthogonal to the entire space, and any such vector in the radical satisfies B(w, w) = 0, hence is isotropic for q. Nondegeneracy of q (i.e., \mathrm{rad}(B) = \{0\}) ensures that isotropic vectors, when present, reflect genuine geometric splitting rather than degeneracy.^[7]^[8] In fields of characteristic 2, the division by 2 is unavailable, so a quadratic form q does not uniquely determine a symmetric bilinear form via the above formula; instead, the associated form is B(u, v) = q(u + v) + q(u) + q(v), which is symmetric but satisfies q(v) \neq B(v, v) in general. Analyzing isotropy then requires additional structure, such as the associated alternating form or the Arf invariant, to distinguish quadratic forms beyond their bilinear polarizations.^[7]

Properties

Existence Conditions

The existence of isotropic quadratic forms depends fundamentally on the properties of the underlying field and the dimension of the vector space on which the form is defined. Over an algebraically closed field, such as the complex numbers, every non-degenerate quadratic form of dimension at least 2 is isotropic. This follows from the fact that such forms can be diagonalized, and the equation q(x_1, x_2) = a x_1^2 + b x_2^2 = 0 always admits a non-trivial solution, for instance by setting x_2 = 1 and solving for x_1 = \sqrt{-b/a} when a, b \neq 0, leveraging the algebraic closure. Over finite fields, the Chevalley-Warning theorem provides a key criterion: every non-degenerate quadratic form of dimension at least 3 is isotropic. The theorem states that for a polynomial of degree d in n variables over a finite field \mathbb{F}_q, if n > d, then the number of solutions to the equation is congruent to 0 modulo p (where q = p^k), implying non-trivial zeros for the homogeneous quadratic case in dimension 3 or higher. This result, originally proved by Chevalley in 1935 and strengthened by Warning in the same year, ensures isotropy without requiring additional field properties beyond finiteness. For dimension 2, binary forms may be anisotropic, such as certain norm forms. Over the real numbers, a quadratic form is isotropic if and only if it is indefinite, meaning its signature includes both positive and negative eigenvalues, as classified by Sylvester's law of inertia. This law asserts that every real symmetric matrix is congruent to a diagonal matrix with p entries of +1, n of -1, and the rest zero, where p + n is the rank, and the form is indefinite if p > 0 and n > 0. Such indefinite forms admit non-trivial real zeros in dimensions at least 2 (e.g., x^2 - y^2 = 0), while definite forms are anisotropic in all dimensions. Positive semi-definite forms with zeros are degenerate. More generally, over non-formally real fields F (i.e., fields where -1 is a sum of squares), the u-invariant u(F) is finite, so every non-degenerate quadratic form of dimension greater than u(F) is isotropic.^[9] For fields where -1 is a square (level s(F)=1), this holds for dimensions at least 3. This contrasts with formally real fields like the reals where anisotropic forms can exist indefinitely. The result ties into broader local-global principles, such as the Hasse-Minkowski theorem over number fields, where isotropy over all completions implies global isotropy. In most fields of characteristic not 2, the minimal dimension for possible isotropy is 2, reflecting the existence of hyperbolic planes.

Dimension Constraints

A quadratic form q on a vector space V over a field F of characteristic not 2 is isotropic if there exists a non-zero vector v \in V such that q(v) = 0. For such forms, the dimension of V, denoted \dim V = n, must satisfy n \geq 2, as the existence of a non-trivial isotropic vector implies the presence of a hyperbolic plane—a 2-dimensional subspace where q takes both positive and negative values relative to the associated bilinear form—within V.^[4] Witt's decomposition theorem states that any quadratic space (V, q) decomposes orthogonally as V = H^{\oplus \nu} \oplus A, where H is a hyperbolic plane, \nu is the number of such planes, and A is an anisotropic kernel (a space with no non-trivial isotropic vectors). If q is isotropic, then \nu \geq 1. The Witt index \nu(q), defined as the dimension of a maximal isotropic subspace of V, satisfies $0 \leq \nu(q) \leq \lfloor n/2 \rfloor, with q isotropic if and only if \nu(q) > 0. This upper bound arises because any isotropic subspace pairs with its orthogonal complement under the bilinear form, limiting the maximum size to half the dimension, rounded down.^[4] In even dimensions n = 2m, an isotropic quadratic form splits into \nu hyperbolic planes plus an anisotropic kernel of dimension $2m - 2\nu. The maximum \nu = m occurs when the form is hyperbolic, meaning the anisotropic kernel is trivial (dimension 0). For odd dimensions n = 2m + 1, the anisotropic kernel has at least dimension 1, so \nu \leq m.^[4] Anisotropic quadratic forms, by contrast, have \nu(q) = 0 and thus impose stricter dimensional limits depending on the field F. The u-invariant u(F) is the supremum of dimensions of anisotropic forms over F. Over the rational numbers \mathbb{Q}, u(\mathbb{Q}) = 4, so anisotropic forms exist only in dimensions at most 4; for example, the form x^2 + y^2 + z^2 + w^2 is anisotropic. Over the real numbers \mathbb{R}, u(\mathbb{R}) = \infty, allowing anisotropic (positive definite) forms in arbitrarily high dimensions, such as the standard sum of squares in any n. These constraints highlight how isotropy becomes inevitable in high dimensions over fields with finite u-invariant.^[10]

Constructions and Examples

Hyperbolic Plane

The hyperbolic plane H is a 2-dimensional quadratic space over a field K of characteristic not 2, equipped with a basis \{e, f\} such that the quadratic form satisfies q(e) = q(f) = 0 and the associated symmetric bilinear form B satisfies B(e, f) = 1.^[11] This configuration ensures that H is isotropic, as both basis vectors are null (isotropic) vectors, and the space is non-degenerate since the Gram matrix of B has non-zero determinant.^[11] With respect to the basis \{e, f\}, the Gram matrix of the bilinear form B is

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},

which has determinant -1 and reflects the symplectic structure underlying the form.^[11] In contrast, attempting to represent H in diagonal form as \operatorname{diag}(0, 0) results in a degenerate bilinear form, as its matrix would have determinant 0 and fail to capture the non-degeneracy.^[7] Equivalently, H can be presented via the quadratic form q(x, y) = xy on K^2, where the associated B((x_1, y_1), (x_2, y_2)) = x_1 y_2 + y_1 x_2.^[4] Over fields of characteristic not 2, all non-degenerate 2-dimensional isotropic quadratic spaces are isomorphic to the hyperbolic plane.^[7] This uniqueness up to isomorphism follows from the fact that any such space admits a hyperbolic pair—a pair of isotropic vectors u, v with B(u, v) = 1—spanning the space, and the isometry class is determined by this pairing.^[11] Geometrically, the hyperbolic plane embodies an indefinite metric with two orthogonal null directions, providing a model for light-like separations and serving as a basic building block in the 2-dimensional slice of Minkowski space central to special relativity.

Split Quadratic Spaces

A split quadratic space, also known as a hyperbolic quadratic space, is a quadratic space (V, q) over a field k of characteristic not 2 that is isometric to the orthogonal direct sum of n copies of the hyperbolic plane H, denoted V \cong H^{\oplus n}, where \dim V = 2n and the Witt index of V is n.^[4] This structure ensures maximal isotropy, meaning the largest totally isotropic subspace has dimension n, which is half the dimension of V. In a non-degenerate split quadratic space, the radical is trivial, as the quadratic form has no kernel beyond the zero vector.^[4] The orthogonal group O(V) \cong O(2n, k) acts transitively on the set of totally isotropic subspaces of any dimension d \leq n.^[4] Over any field k, a split quadratic space of rank $2n admits a basis in which the Gram matrix of the associated symmetric bilinear form is block-diagonal, consisting of n copies of the $2 \times 2 matrix \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.^[4] For example, over the rational numbers \mathbb{[Q](/page/Q)}, the quadratic form q(x_1, y_1, \dots, x_n, y_n) = \sum_{i=1}^n x_i y_i defines a split quadratic space of dimension $2n, as it decomposes into n hyperbolic planes.^[4] This contrasts with anisotropic forms over \mathbb{[Q](/page/Q)}, such as x^2 + y^2, which have Witt index 0 and admit no nontrivial isotropic vectors.^[4]

Classifications

Relation to Quadratic Form Classification

The classification of quadratic forms over a field K relies on a set of invariants that capture their isomorphism classes, including the dimension, the discriminant (a class in K^\times / K^{\times 2}), the Hasse invariant (or more generally the Clifford invariant), and the Witt index, which measures the extent of isotropy.^[12] The Witt index i_W(V) of a quadratic space V is the dimension of a maximal totally isotropic subspace, and a form is isotropic if and only if i_W(V) > 0.^[4] Isotropy thus plays a pivotal role in this classification, as forms with positive Witt index are distinguished from anisotropic ones (where i_W(V) = 0) and are further parameterized by the index value, which determines the number of hyperbolic planes in their Witt decomposition.^[12] For isotropic quadratic forms, the classification simplifies significantly due to Witt's cancellation theorem, which states that if two quadratic spaces U_1 \oplus H \cong U_2 \oplus H, where H is a hyperbolic plane, then U_1 \cong U_2.^[4] Consequently, an isotropic form is uniquely determined up to isomorphism by its anisotropic kernel—the unique anisotropic quadratic space that complements the hyperbolic part in the Witt decomposition. This kernel inherits the invariants of the original form adjusted for the hyperbolic summands, allowing classification to reduce to that of the smaller anisotropic component.^[12] In the context of number fields like \mathbb{Q}, isotropy of quadratic forms is governed by the local-global principle via the Hasse-Minkowski theorem, which asserts that a quadratic form over \mathbb{Q} is isotropic if and only if it is isotropic over \mathbb{R} and over every \mathbb{Q}_p for primes p.^[13] Local isotropy at all places thus implies global isotropy, with the Witt index over \mathbb{Q} matching the minimum of the local indices under these conditions.^[12] Anisotropic forms, by contrast, require separate classification methods, such as the signature (number of positive and negative eigenvalues) over \mathbb{R}, without reduction to hyperbolic components. Isotropic forms, however, universally decompose as a direct sum of a hyperbolic space and an anisotropic kernel, streamlining their classification across fields. Split quadratic spaces represent the extreme case of maximal isotropy, where the Witt index equals half the dimension.^[12]

Over Fields of Characteristic Not 2

Over fields of characteristic not 2, the classification of isotropic quadratic forms relies on a set of standard invariants that distinguish non-degenerate quadratic spaces up to isometry. The discriminant d(q) of a quadratic form q on an n-dimensional space is defined as d(q) = (-1)^{n(n-1)/2} \det(B), where B is the Gram matrix of the associated symmetric bilinear form, taken modulo squares in F^\times / F^{\times 2}. This invariant captures the "oriented" determinant and is essential for local-global principles. The Clifford invariant, residing in the 2-torsion of the Brauer group \mathrm{Br}_2(F), arises from the class of the Clifford algebra and is particularly relevant for even-dimensional forms, providing cohomological information about the form's structure. The Witt index \nu(q), defined as the dimension of a maximal totally isotropic subspace, measures the hyperbolic part of the form; a form is isotropic if and only if \nu(q) \geq 1.^[14] A fundamental classification theorem states that two non-degenerate quadratic forms over a number field F (such as \mathbb{Q}) are isometric if and only if they have the same dimension, the same discriminant, and the same Hasse-Witt invariants at every place of F. The Hasse-Witt invariant, also known as the Hasse invariant, is a local symbol in \{ \pm 1 \} computed via the Hilbert symbol for diagonalizations, reflecting the form's behavior over completions. For isotropy specifically, the condition \nu(q) \geq 1 aligns with the form admitting a non-trivial zero, and the local-global principle (Hasse-Minkowski theorem) ensures a form over F is isotropic if and only if it is isotropic over every local completion. This theorem underpins the Witt decomposition, where isotropic forms split into an anisotropic kernel and hyperbolic summands, with the number of hyperbolic planes determined by the Witt index.^[14]^[4] Over local fields such as the p-adic numbers \mathbb{Q}_p with p odd, every non-degenerate quadratic form of dimension at least 3 is isotropic, as guaranteed by properties of the Hilbert symbol and diagonalization. For p=2 over \mathbb{Q}_2, the situation is more nuanced: forms of dimension at least 5 are isotropic, but there exist anisotropic forms in dimensions 3 and 4, such as the 3-dimensional form x^2 + xy + 3y^2 + 3z^2 or the unique (up to isometry) 4-dimensional anisotropic form \langle 1, -u, -\pi, u\pi \rangle, where u is a non-square unit and \pi = 2 is the uniformizer. These exceptions highlight the role of the dyadic case in complicating local isotropy criteria, though the classification still proceeds via dimension, discriminant, and Hasse invariant.^[15]^[14] A concrete example over the rationals \mathbb{[Q](/page/Q)} is the ternary form q(x,y,z) = x^2 + y^2 - z^2, which is isotropic because it represents zero non-trivially; for instance, q(3,4,5) = 0, corresponding to primitive Pythagorean triples generated by the parametrization x = m^2 - n^2, y = 2mn, z = m^2 + n^2 for coprime integers m > n > 0 of opposite parity. This form has dimension 3, discriminant -1 modulo squares, and Witt index 1, illustrating how isotropy over \mathbb{[Q](/page/Q)} follows from local solubility everywhere by the Hasse-Minkowski theorem.^[14]

Advanced Contexts

Over Fields of Characteristic 2

Over fields of characteristic 2, the standard association between a quadratic form q: V \to F and a symmetric bilinear form breaks down, as the polarization identity yields an alternating bilinear form instead. Specifically, the associated bilinear form is given by b(u,v) = q(u+v) + q(u) + q(v), which is bilinear and alternating (b(v,v) = 0 for all v \in V), though it coincides with a symmetric form in this characteristic.^[16]^[7] The notion of isotropy adapts accordingly: a quadratic form q is isotropic if there exists a nonzero v \in V such that q(v) = 0. In this setting, the radical of b, defined as \mathrm{Rad}(b) = \{v \in V \mid b(v,w) = 0 \ \forall w \in V\}, may exceed the isotropic cone \{v \in V \mid q(v) = 0\}, and q is regular (nonsingular) if \mathrm{Rad}(b) = 0, which requires \dim V to be even.^[7]^[17] The classification of such forms relies on the Arf invariant, an element of F / \wp(F) where \wp(x) = x^2 + x, which distinguishes equivalence classes of regular quadratic forms and captures the "type" of the anisotropic kernel.^[7]^[18] Regular quadratic forms over fields of characteristic 2 are classified as quasi-split (isotropic, containing a hyperbolic plane) or anisotropic, with the former admitting a Witt decomposition into hyperbolic and anisotropic summands. Over the prime field \mathbb{F}_2, nonsingular forms exist only in even dimensions; in dimension 2, there are exactly two isomorphism classes—one isotropic (the hyperbolic plane) and one anisotropic—while dimension 1 admits only degenerate forms.^[7]^[19] A key result on existence and classification is the following: over any finite field of characteristic 2, every regular quadratic form of dimension at least 3 is isotropic, and all regular forms are classified up to isomorphism by their dimension and Arf invariant (with the latter taking two possible values).^[19]^[20] This follows from the Chevalley-Warning theorem applied to the projective variety defined by the quadratic equation, ensuring nontrivial zeros in sufficiently high dimensions.^[19] The Arf-Brown invariant provides a refined tool for classifying the Witt group in this context, incorporating Clifford algebra structure for higher invariants.^[18] The Witt index, measuring the maximal dimension of totally isotropic subspaces, remains applicable but requires adjustment in its computation due to the alternating bilinear structure.^[17]

Witt Decomposition Applications

The Witt decomposition theorem asserts that every non-degenerate quadratic space (V, q) over a field of characteristic not equal to 2 admits a unique orthogonal direct sum decomposition V = V_h \oplus V_a, where V_h is a hyperbolic space (the maximal isotropic subspace, consisting of a direct sum of hyperbolic planes) and V_a is an anisotropic quadratic space (the anisotropic kernel, with no non-trivial isotropic vectors).^[21] This decomposition captures the isotropic structure of the form by separating the "hyperbolic" component, which determines the extent of isotropy, from the "rigid" anisotropic remainder.^[14] The decomposition can be constructed algorithmically by iteratively identifying isotropic vectors and extending them to hyperbolic planes via the Witt extension theorem, which guarantees that any isometry between subspaces extends to the full space, allowing subtraction of these planes until only the anisotropic kernel remains.^[21] This process is finite, as each step reduces the dimension by 2, and the uniqueness ensures that the number of hyperbolic planes (the Witt index) and the isometry class of V_a are invariants of the quadratic space.^[14] Applications of the Witt decomposition include computing the Witt index, defined as half the dimension of V_h, which quantifies the maximal dimension of isotropic subspaces and plays a central role in the representation theory of quadratic forms.^[21] It is fundamental in index theory for orthogonal groups, where the index relates to the connectivity of the special orthogonal group and the topology of associated Grassmannians.^[22] Additionally, the decomposition links to spinor norms, which classify similitudes between quadratic spaces and arise in the study of the spinor genus in integral quadratic forms.^[14] For example, consider an indefinite quadratic form over the reals in dimension 4 with signature (3,1), such as q = x_1^2 + x_2^2 + x_3^2 - x_4^2. Its Witt decomposition is H \oplus \langle 1,1 \rangle, where H is a hyperbolic plane and \langle 1,1 \rangle represents the anisotropic positive definite binary form x^2 + y^2, yielding a Witt index of 1.^[21]

References

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Every form 𝑞 is isomorphic to 𝑞𝑎𝑛 ⟂ 𝑘ℍ, where 𝑞𝑎𝑛 is an anisotropic form and 𝑘 ≥ 0. The integer 𝑤(𝑞) = 𝑘 is called the Witt index of 𝑞.
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