III.73 Quadratic Forms

Ben Green

A quadratic form is a homogeneous polynomial of degree 2 in some finite set of unknowns x_l, x₂,..., x_n: an example is q(x_l, x₂, x₃) = - 3x₁x₂ + 4. Here, the coefficients 1, -3, and 4 are integers, but the idea generalizes straightforwardly from to any ring R. Since linear functions are undeniably important and 2 is the next positive integer after 1, one might expect quadratic forms to be important as well, and indeed they are, in many different branches of mathematics, including linear algebra itself.

Here are two theorems about quadratic forms.

Theorem 1. If x, y, and z are three points in ^d, then the distances between them satisfy the triangle inequality

|x - z| ≤ |x - y| + |y - z|.

Theorem 2. An odd prime p can be written as the sum of two squares if and only if it leaves remainder 1 on division by 4.

It is not at first sight clear why theorem 1 has anything to do with quadratic forms. The reason is that the square of the Euclidean distance

is a quadratic form over the real numbers (here, the x_i are the coordinates of x). This form is derived from the inner product

〈x, y〉 = x₁y₁ + ··· + x_dy_d

by taking |x|² to be 〈x, x〉. The inner product satisfies the relations

(i) 〈x, x〉 ≥ 0 for all x ∈ ^d, with equality if and only if x = 0.

(ii) 〈x, y + z〉 = 〈x, y〉 + 〈x, z〉 for all x, y, z ∈ ^d.

(iii) 〈x, y〉 = 〈x, y) = 〈x, y〉 for all ∈ and x, y ∈ ^d.

(iv) 〈x, y〉 = 〈y, x〉 for all x, y ∈ ^d.

More generally, any function (x, y) that satisfies these relations is called an inner product. The triangle inequality is a consequence of arguably the most important inequality in mathematics, the CAUCHY-SCHWARZ INEQUALITY [V.19]

|〈x, y〉 | ≤ |x| |y |.

Not all quadratic forms on ^d come from inner products, but they do all come from symmetric bilinear forms g : ^d × ^d →, . These are functions of two variables that satisfy all the axioms of an inner product except possibly (i), the positivity criterion. Given a quadratic form q(x) = g(x, x), one may recover g using the polarization identity

g(x, y) = (q(x + y) - q(x) - q(y)).

This correspondence between quadratic forms and symmetric bilinear forms works just as well when is replaced by any field k, except that there are some serious technical issues when k has characteristic two (due to the presence of the fraction in the above formula). In linear algebra one often defines quadratic forms by first discussing symmetric bilinear forms. The advantage of this more abstract approach over the concrete definition we gave at the beginning is that it is not necessary to specify a basis for ^d.

If one makes a good choice of basis, then the quadratic form can be made to look particularly pleasant: we may always choose a basis in such a way that

for some s and t satisfying 0 ≤ s ≤ t ≤ d. Here x₁, . . . , x_t are the coefficients of x with respect to the basis we have carefully chosen. The quantity s - t is called the signature of the form. When s = d (as is the case for the form defining the Euclidean distance) the form is said to be positive definite. Forms that are not positive definite occur very commonly. For example, the form x² + y² + z² - t² is used to define MTNKOWSKI SPACE [I.3 §6.8], which plays a key role in special relativity.

We turn now to examples of quadratic forms in number theory, beginning with two very famous theorems about quadratic forms over the integers . The first is theorem 2, mentioned at the start of the article. It is due to FERMAT [VI.12]. There are many related results for other binary quadratic forms such as x² + 2y² and x² + 3y². In general, however, the question of which primes are represented by x² + ny² is extremely subtle and interesting, and leads one to CLASS FIELD THEORY [V.28].

In 1770 LAGRANGE [VI.22] showed that every number n can be written as a sum of four squares. In fact, the number of such representations of n, r₄(n), is given by the formula

This formula can be explained using the theory of MODULAR FORMS [III.59], one of the most important topics in number theory. Indeed, the generating series

is a theta series, as a result of which it satisfies certain transformations that identify it as a modular form.

A remarkable theorem of Conway and Schneeberger states that if a quadratic form with a1, . . . , a₄ ∈ represents all the positive integers less than or equal to 15, then it represents all positive integers. RAMANUJAN [VI.82] listed fifty-five such forms; actually, one of his forms did not represent 15, but the remaining fifty-four forms constitute the complete list. For example, every positive integer can be written as .

Quadratic forms in three variables are more difficult to treat. GAUSS [VI.26] proved that if and only if n does not have the form 4^t(8k + 7) for integers t and k. It is still not known exactly which integers can be written as (this is known as Ramanujan′s ternary form).

From the point of view of prime number theory, quadratic forms in one variable are the hardest to understand. For example, are there infinitely many primes of the form x² + 1?

Let us mention one final topic, where quadratic forms over R are studied but where the unknowns x₁, . . . , x_n are replaced by integers. In particular, let us mention a beautiful result of Margulis, which confirmed a conjecture of Oppenheim. One instance of the result is the following: for any > 0, one may find integers x₁, x₂, and x₃ such that

The proof uses techniques from ERGODIC THEORY [V.9], which in related contexts are proving very influential at the forefront of research today. No explicit bounds are known on how large x₁, x₂, and x₃ need to be.