Square shape

A quadratic form is a function in mathematics that behaves like the quadratic function in some respects . The best known example is the square of the amount of a vector. Square shapes appear in many areas of mathematics. In geometry they are used to introduce metrics , in elementary geometry to describe conic sections . If, for example, they are considered over the rational or whole numbers, they are also a classic subject of number theory ${\displaystyle x\mapsto x^{2}}$, in which one asks about the numbers that can be represented by a square shape. In the following, number theoretic aspects are mainly considered.

motivation

A (real) vector space with scalar product can be made into a normalized space by defining the norm of a vector as an induced norm . The square root used here interferes insofar as one can generalize to more general bilinear forms and other basic bodies if one looks at the mapping instead . Since a vector space is determined by the fact that vectors can be added and scaled with elements of the basic body, it must be investigated how the mapping behaves here. The following relationships are found: ${\displaystyle V}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle x}$ ${\displaystyle \|x\|:={\sqrt {\langle x,x\rangle }}}$ ${\displaystyle q\colon x\mapsto \langle x,x\rangle }$ ${\displaystyle K}$ ${\displaystyle q}$

${\displaystyle {\begin{array}{ll}q(ax)=a^{2}q(x)&\mathrm {f{\ddot {u}}r\ alle} \quad a\in K{\text{ und }}x\in V\\q(x+y)+q(x-y)=2q(x)+2q(y)&\mathrm {f{\ddot {u}}r\ alle} \quad x,y\in V\end{array}}}$

Images that meet the above conditions can also be viewed without deriving from a bilinear form. On top of that one can generalize from vector spaces over a field to modules over a commutative ring with one element . Often one examines the ring of whole numbers as well as the module , in particular . ${\displaystyle q\colon V\to K}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} ^{n}}$ ${\displaystyle \mathbb {Z} ^{2}}$

Definitions

Square form in n indeterminates

A quadratic form (in indeterminate) over a commutative ring with one element is a homogeneous polynomial of degree  2 in indeterminate with coefficients in . ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle A}$

The term form was coined by Legendre .

Special cases

• For one speaks of binary quadratic forms . A binary quadratic form is therefore a polynomial of the form with . ${\displaystyle n=2}$ ${\displaystyle aX^{2}+bXY+cY^{2}}$ ${\displaystyle a,b,c\in A}$

• For one speaks of ternary square forms . A ternary quadratic form is therefore a polynomial of the form with . ${\displaystyle n=3}$ ${\displaystyle aX^{2}+bXY+cXZ+dY^{2}+eYZ+fZ^{2}}$ ${\displaystyle a,\dotsc ,f\in A}$

Square shape on modules

More generally, the term square shape is defined for any A- module as follows: A square shape on is a map with the following properties: ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle q\colon M\to A}$

• For everyone and applies . ${\displaystyle a\in A}$ ${\displaystyle x\in M}$ ${\displaystyle q(ax)=a^{2}q(x)}$
• The figure defined by linear in both arguments, ie a bilinear form on . It is automatically symmetrical, so it applies . It is called the associated symmetrical bilinear form . ${\displaystyle b\colon M\times M\to A}$ ${\displaystyle b(x,y):=q(x+y)-q(x)-q(y)}$ ${\displaystyle M}$ ${\displaystyle b(x,y)=b(y,x)}$ ${\displaystyle q}$

A square shape in the above sense is therefore a square shape on the module . ${\displaystyle A^{n}}$

Square module

A quadratic module is a pair consisting of an A module and a square shape on . ${\displaystyle (M,q)}$ ${\displaystyle M}$ ${\displaystyle q}$ ${\displaystyle M}$

It denotes the associated symmetrical bilinear form . Then two elements are called -orthogonal or -orthogonal , if holds. ${\displaystyle b}$ ${\displaystyle q}$ ${\displaystyle x,y\in M}$ ${\displaystyle q}$ ${\displaystyle b}$ ${\displaystyle b(x,y)=0}$

Square space

A square space is a square module , where is a vector space . The ring above which is defined, that is a body . ${\displaystyle (V,q)}$ ${\displaystyle V}$ ${\displaystyle V}$

Algebraic requirements

In the following it is assumed that it is invertible in the ring . This is particularly true for bodies of characteristic different from 2 as the real or complex numbers. ${\displaystyle 2}$ ${\displaystyle A}$

If one assigns a square shape , the triangular matrix with , otherwise 0) to, it can be as as or interpreted. This initially results: ${\displaystyle \textstyle q(x)=\sum _{1\leqslant i\leqslant j\leqslant n}q_{ij}x_{i}x_{j}}$ ${\displaystyle Q=(q_{ij}}$ ${\displaystyle i\leqslant j}$ ${\displaystyle q(x)}$ ${\displaystyle x^{T}Qx}$ ${\displaystyle x^{T}Q^{T}x}$

Relation to symmetrical bilinear forms

There is a clear correspondence between quadratic forms in indefinite and symmetric bilinear forms on : ${\displaystyle n}$ ${\displaystyle A^{n}}$

A symmetrical bilinear shape is obtained for a square shape by polarization ${\displaystyle q}$ ${\displaystyle B}$

${\displaystyle B(x,y)={\frac {1}{2}}\left(q(x+y)-q(x)-q(y)\right).}$

The opposite is true

${\displaystyle q(x)=B(x,x).}$

From a formal point of view, this construction initially only provides a polynomial function; but you actually get a polynomial by representing the bilinear form by a matrix or by extending it to any - algebras . ${\displaystyle A}$

Equivalence of forms

If there is a - row matrix, then the substitution gives a new square shape . If it can be inverted, the old form can also be recovered from the new form. Overall, such a matrix group enables the introduction of an equivalence relation on the set of all quadratic forms. We are talking about -equivalent forms here (also note the concluding remark on 4). ${\displaystyle S}$ ${\displaystyle n}$ ${\displaystyle y=Sx}$ ${\displaystyle y^{T}(S^{T}QS)y}$ ${\displaystyle S}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$

Definiteness

For real or rational forms, one can use the corresponding matrix criteria for ( definiteness ) to obtain statements about whether the value range of the form assumes only positive or only negative values, or whether such a restriction does not apply. Accordingly, the form is called positive definite, negative definitive or indefinite. If the range of values ​​for definition values ​​not equal to zero only assumes positive or negative values ​​as well as zero, the form is called positive or negative semidefinite. ${\displaystyle Q+Q^{T}}$ ${\displaystyle \mathbb {R} ^{n}}$

Examples / classification

Quadratic forms over the real numbers

Let it be a - vector space . According to Sylvester's law of inertia , every square shape can be diagonalized; H. there exists a base of such that ${\displaystyle V}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle q\colon V\to \mathbb {R} }$ ${\displaystyle e_{1},\dotsc ,e_{n}}$ ${\displaystyle V}$

${\displaystyle q(\lambda _{1}e_{1}+\dotsb +\lambda _{n}e_{n})=\lambda _{1}^{2}+\dotsb +\lambda _{a}^{2}-\lambda _{a+1}^{2}-\dotsb -\lambda _{a+b}^{2}}$

for certain with applies. The isomorphism class of a square shape is determined by its rank and signature . ${\displaystyle a,b}$ ${\displaystyle a+b\leq n}$ ${\displaystyle a+b}$ ${\displaystyle a-b}$

Square forms over number fields

Square shapes over were classified by Minkowski . Hasse later generalized this to a classification of quadratic forms over number fields . In particular, two quadratic forms are isomorphic if and only if all of their completions (real, complex and p-adic) are isomorphic, see Hasse-Minkowski's theorem . ${\displaystyle \mathbb {Q} }$

Square shapes over the whole numbers

It is said that two positive-definite quadratic forms have the same sex if one gets isomorphic quadratic forms for all by expanding with scalars to (i.e. tensor product with ) isomorphic quadratic forms . The number of isomorphism classes of the same sex can be determined using the Smith-Minkowski Siegel mass formula . ${\displaystyle (V,q),(V^{\prime },q^{\prime })}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$

Elementary number theory

There are a multitude of results for the question of whether a given integer square form with any integer arguments can assume a given value ("represents or represents a value"). Viewed in isolation, these results are of course often anecdotal. Note, however, that

• ${\displaystyle \operatorname {SL} _{n}(\mathbb {Z} )}$, the group of -series, integer matrices of the determinant 1, and ${\displaystyle n}$
• ${\displaystyle \operatorname {GL} _{n}(\mathbb {Z} )}$, the group of -series, integer matrices of the determinant ± 1, ${\displaystyle n}$

If both the lattice and the set of coprime numbers are bijectively mapped onto itself, the following results stand for whole families of equivalent forms. ${\displaystyle \mathbb {Z} ^{n}}$ ${\displaystyle \mathbb {Z} ^{n}}$

The following topics are prominent, for example

Square numbers of the shape ${\displaystyle x^{2}+y^{2}}$

The integer solutions of the equation are called Pythagorean numbers. The best known solution to this problem is . This is the smallest of an infinite number of solutions. ${\displaystyle x^{2}+y^{2}=z^{2}}$ ${\displaystyle 3^{2}+4^{2}=5^{2}}$

More than the usual parametric description of all solutions ( Pythagorean triple ) can be found in the literature.

Numbers of the form ${\displaystyle w^{2}+x^{2}+y^{2}+z^{2}}$

The first known case of a square shape representing all natural numbers. ( Lagrangian theorem or four-squares theorem )

A proof and further information on the subject of quadratic forms, which represent all natural numbers, via the 15-Theorem .

integer solutions of the equation ${\displaystyle ax^{2}+by^{2}+cz^{2}=0}$

( integer, square-free, coprime pairs, not all of the same sign ). ${\displaystyle a,b,c}$

A non-trivial solution exists if and only if , and are quadratic remainders in the respective module. This is a result of Legendre (for the notation see congruence (number theory) ). ${\displaystyle -ab{\pmod {c}}}$ ${\displaystyle -bc{\pmod {a}}}$ ${\displaystyle -ca{\pmod {b}}}$

Prime numbers of the form ${\displaystyle x^{2}+y^{2}}$

These are exactly 2 as well as the prime numbers . The observation is of particular historical importance, it goes back to Fermat . ${\displaystyle \equiv 1{\pmod {4}}}$

A modern proof, almost the mother of all proofs, in the book of proofs chapter 4.

Prime numbers of the form ${\displaystyle x^{2}+xy+y^{2}}$

These are exactly the 3 as well as the prime numbers that are. ${\displaystyle \equiv 1{\pmod {3}}}$

Prime numbers of the form ${\displaystyle x^{2}+ny^{2}}$

Cox's book deals with this question.

If two square shapes emerge from each other by using a matrix , then an integer can be represented as the value of one square shape if and only if it can be represented as the value of the other square shape: this follows directly from the definition . From the point of view of number theory, the forms and are therefore equivalent and the question arises of finding the simplest possible system of representatives for the set of quadratic forms in variables modulo the effect of . For quadratic forms in 2 variables, this problem was discussed by Gauss in Chapter 5 of " Disquisitiones Arithmeticae " (the main part of the book with almost 260 pages). ${\displaystyle A\in \operatorname {GL} _{n}(\mathbb {Z} )}$ ${\displaystyle (Aq)(x_{1},\dotsc ,x_{n})=q(A^{-1}x_{1},\dotsc ,A^{-1}x_{n})}$ ${\displaystyle q}$ ${\displaystyle Aq}$ ${\displaystyle n}$ ${\displaystyle \operatorname {GL} (n,\mathbb {Z} )}$

In the case of positively definite quadratic forms the problem in today's language is to find a fundamental domain for the effect of on the symmetric space (the space of positively definite quadratic forms in variables). ${\displaystyle \operatorname {GL} (n,\mathbb {Z} )}$ ${\displaystyle \operatorname {GL} (n,\mathbb {R} )/O(n)}$ ${\displaystyle n}$

Fundamental domain for the action of SL (2, ℤ) on the hyperbolic level.

For the space of the positive definite binary quadratic forms can be identified with the hyperbolic plane . The picture on the right shows a decomposition of the hyperbolic level into fundamental areas for the effect of . Such a fundamental domain (e.g. the one hatched in gray in the picture) thus provides a representative system of binary square shapes, so that every other positively definite binary square shape is equivalent to a shape from the representative system and in particular represents the same whole numbers. ${\displaystyle n=2}$ ${\displaystyle \operatorname {GL} (2,\mathbb {R} )/O(2)}$ ${\displaystyle \operatorname {GL} (2,\mathbb {Z} )}$

Related questions, albeit outside the realm of quadratic forms, are subjects such as Fermat's theorem and the Waring problem .

Related terms

The ( projective ) set of zeros of a square shape is called a quadric .

literature

• Martin Kneser , Rudolf Scharlau: Square forms . Springer Verlag, 2002, ISBN 3-540-64650-7 (Lectures by Kneser in the 1970s and 1980s in Göttingen, re-edited by Scharlau)
• Winfried Scharlau : Quadratic and Hermitian Forms . Basic Teachings of Mathematical Sciences, Volume 270.Springer Verlag, 1985
• John Milnor , Dale Husemöller: Symmetric bilinear forms . Springer Verlag, 1973

Web links

• Springer Encyclopaedia of Mathematics
  • Quadratic shape
  • binary quadratic form

Individual evidence

1. ↑ ^{a b} David Cox: Primes of the form ${\displaystyle x^{2}+ny^{2}}$ . Wiley & Sons (1997), 40.
2. ^ Roger C. Alperin: The modular tree of Pythagorus . (PDF)
3. ↑ Dan Romik: The dynamics of Pythagorean triples . (PDF) with a whole range of other references.
4. ↑ Kenneth Ireland, Michael Rosen: A Classical Introduction to Modern Number Theory . Springer-Verlag, 1982, section 17.7.
5. ↑ Kenneth Ireland, Michael Rosen: A Classical Introduction to Modern Number Theory . Springer-Verlag, 1982, section 17.3.1.
6. ^ Martin Aigner, Günter M. Ziegler : Proofs from the Book . Springer-Verlag, 2000
7. ^ GH Hardy, EM Wright: An Introduction to the Theory of Numbers . 4th edition. Oxford University Press, Oxford 1975, ISBN 0-19-853310-1 : Theorem 366, p. 299; Theorem 254, p. 221