From Pythagorean triples to Gauss, quadratic forms connect ancient puzzles and modern maths
Which whole numbers can be written as the sum of two squares? That question, answered by Fermat, belongs to the theory of quadratic forms, polynomials where every term has degree two. The same simple expressions describe distance in space, the bell curve of statistics and the shape of four-dimensional manifolds.
A quadratic form is a polynomial whose terms are all of degree two, such as x squared plus a multiple of xy plus y squared. With one, two or three variables they are called unary, binary and ternary. They differ from quadratic equations, which involve a single variable and may include lower powers. The most familiar example is the squared distance of a point from the origin, x squared plus y squared plus z squared. Over the real numbers, a form is definite if it vanishes only when every variable is zero, and isotropic otherwise.
The ideas are old. Pythagorean triples were already known more than three thousand years ago, in the second millennium BCE. Brahmagupta, writing in India in 628, studied equations like x squared minus n times y squared equals c, including the case now called Pell's equation, and found a way to solve it; in Europe the problem later occupied Brouncker, Euler and Lagrange. In 1801 Carl Friedrich Gauss devoted much of his Disquisitiones Arithmeticae to a full theory of binary quadratic forms over the integers.
Linear algebra gives them a clean structure. Every square matrix defines a quadratic form, and over the real numbers each form corresponds to exactly one symmetric matrix. A central problem is classifying forms under changes of variables, and Jacobi proved that any real form can be rotated into a diagonal version, a simple weighted sum of squares. Geometrically, real forms in three variables can be pictured as conic sections.
How the theory behaves depends heavily on what numbers the coefficients come from. With real or complex coefficients, forms underpin analytic geometry and most applications. Over general fields they have an algebraic theory, and over the integers or p-adic integers an arithmetic one tied to quadratic fields, continued fractions and modular forms. They also turn up in the Riemannian metric of differential geometry, the Killing form of Lie theory, intersection forms of four-dimensional manifolds, and in the exponent of the multivariate normal distribution.
Source: Quadratic form
