Quadratic form

A quadratic form is in mathematics a function that behaves in some aspects such as the quadratic function. The best known example is the square of the magnitude of a vector. Quadratic forms appear in many areas of mathematics. In geometry, they are used to introduce metrics in the elementary geometry for the description of conic sections. They are, however, if, for example, over the rationals or integers considered a classical subject of number theory, in which one asks about the numbers, which can be represented by a quadratic form. Here especially number theoretic aspects are considered in the following.

Motivation

A ( real) vector space with scalar product can be done to a normed space by defining the norm of a vector as an induced norm. The square root used herein interferes in that, if one looks at the image, instead, can be generalized to more general bilinear and other base body. As a vector space is determined by the fact that vectors can be added together and scaled by elements of the body, is to investigate how the image behaves here. One finds the following relations:

• For all and
• For all

Pictures that meet the above conditions, you can also look at without having to originate from a bilinear form. On top of that you can generalize with unit element of vector spaces over a field to modules over a commutative ring. Frequently we examine here the ring of integers and the module, esp.

Definitions

Square shape

A quadratic form ( in indeterminates ) over a commutative ring with unit element is a homogeneous polynomial of degree 2 in indeterminates with coefficients in.

The term was coined by Legendre form.

Special cases

• For one speaks of binary quadratic forms. A binary quadratic form is therefore with a polynomial of the form.

• For one speaks of ternary quadratic forms. A ternary quadratic form is therefore with a polynomial of the form.

Square room

A square space is a pair consisting of a vector space, and a square shape.

Denote the symmetric bilinear form associated to. Then the names of two vectors orthogonal or orthogonal if the following holds.

Algebraic conditions

In the following, it is assumed that in the ring is inverted. This is especially true for fields of characteristic different from 2 as the real or complex numbers.

One arranges a square shape with the triangular matrix, 0 otherwise ), so one can also conceive as or as. It follows from this first:

• Respect to symmetric bilinear forms There is a one-one correspondence between quadratic forms in indefinite and symmetric bilinear forms on: For a quadratic form a symmetric bilinear form obtained by polarization Conversely Formally, this construction initially provides only a polynomial function; but you actually get a polynomial by representing the bilinear form by a matrix or it expands to any algebras.

• Equivalence of forms Where S is a n -row matrix is then obtained by the substitution of a new square shape. If S is invertible, one can recover the old form of the new form again. In total, allowing a matrix group to introduce an equivalence relation on the set of all quadratic forms. We're talking about - equivalent forms (Note also the final remark to 4).

• Definiteness For real or rational forms you can have the appropriate matrix for criteria ( definiteness ) obtain information as to whether the value range of the form takes on only positive or only negative values ​​, or whether such a restriction does not apply. Accordingly, the form is positive definite, negative definite or indefinite called.

Elementary Number Theory

As to whether a given integer quadratic form by any integer arguments can assume a predetermined value ( " represents a value or represents " ), there are a variety of results. When considered in these results naturally have often anecdotal. You, however, noted that

• , The group of the N -row, the integer matrix the determinant of 1, and
• , The group of n -row, integer matrices of determinant ± 1,

Each in bijective mapping both the grid and the amount of prime numbers up, so have the following results for each whole families of equivalent forms.

Prominent example, the following topics

• Squares of the form

• Numbers of the form

• Integral solutions of the equation

• Primes of the form

• Primes of the form

• Primes of the form

If two quadratic forms emerge through the application of a matrix apart, then an integer as the value of a quadratic form can be represented if and only if it can be represented as a value of the other quadratic form: this follows directly from the definition. From the viewpoint of number theory are the forms and therefore equivalent, and it begs the question, the simplest possible set of representatives for the set of quadratic forms in n variables to find modulo the impact of. For quadratic forms in two variables this problem by Gauss in Chapter 5 of " Disquisitiones Arithmeticae " was discussed ( with almost 260 pages of the main part of the book ).

In the case of positive definite quadratic forms, these are in today's language to the problem of finding a fundamental domain for the action of on the symmetric space ( the space of positive definite quadratic forms in n variables).

For n = 2 can be the space of positive definite binary quadratic forms with the hyperbolic plane identify. The accompanying chart shows a decomposition of the hyperbolic plane in fundamental domains for the action of. Such a fundamental domain (eg, the shaded gray in the picture ) so provides a system of representatives of binary quadratic forms, so that any other positive definite binary quadratic form is equivalent to a form of the representative system, and in particular represents the same integers.

Related issues, but outside the range of quadratic forms, are topics such as Fermat's theorem and the Waring problem.

Related terms

The ( projective ) set of zeros of a quadratic form is called a quadric.