Reduction theory quadratic forms
3041 (In Russian) MR0048498.N.
If f a cadeau noel dora x 2 b x y c y 2 displaystyle fax2bxycy2, then important invariants include The discriminant b 2 4.
The most general method is Venkov's method.Over a organiser un jeu concours en entreprise commutative ring with an identity, a homogeneous polynomial in variables with coefficients.Let be the set of matrices over of positive -ary quadratic forms satisfying the equation.Composition means taking 2 quadratic forms of the same discriminant and combining them to create a quadratic form of the same discriminant, it is a generalization of the 2-square identity ( a 2 b 2 ) ( c 2 d 2 ) (.In the field ( ) the problem of equivalence is solved by means of the Hasse invariant.Another way of looking at quadratic forms is the following.On the basis of this algorithm a precise fundamental domain is constructed for.The form is called reducible if is non-empty.The prime examples are the solution of Pell's equation and the representation of integers as sums of two squares.We obtain more pairs that work by switching the values of x displaystyle x and y displaystyle y and/or by changing the sign of one or both of x displaystyle x and y displaystyle.Equivalent quadratic forms represent the same elements.385401 MR08707.10017.If two integral quadratic forms and are equivalent, then there exists an integral transformation, the absolute values of the elements of which are bounded by a constant depending only on and, that takes.
1 Discriminants satisfy 0, 1 ( mod 4 ).
Siegel, "Zur Theorie der quadratischen Formen" Nachr.
Venkov, "The reduction of positive-definite quadratic forms " Izv.Associated with is the point in -dimensional Euclidean space (where called the coefficient space.Every non-degenerate quadratic form is equivalent to a form where is anisotropic; here uniquely determines and the class of the form over, called the anisotropic kernel of (see also Witt decomposition ).Let be the domain of reduction of positive-definite quadratic forms.C) Selling and Charve reduction of a quadratic form.If an integer n pochette cadeau hibou is represented by a form in a class, then it is represented by all other forms in a class.Hasse, "Symmetrische Matrizen im Körper der rationalen Zahlen".Let where is a neighbourhood of in the -dimensional space of -ary quadratic forms over, is the corresponding domain of solutions of the matrix equation (1) over, and and are their volumes.For this reason, the former are called positive definite forms and the latter are negative definite.Die Arithmetik der quadratischen Formen", 12, Teubner (19231925) MR0238661.This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including Shanks's infrastructure, Zagier's reduction algorithm, Conway's topographs, and Bhargava's reinterpretation.Inst., 50 (1975).Buell: Binary Quadratic Forms, Springer, New York 1989 David A Cox, Primes of the form x 2 y 2 displaystyle x2y2, Fermat, class field theory, and complex multiplication Cohen, Henri (1993 A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138, Berlin, New.
Hamburg, 13 (1939).
Also related to the geometric theory of quadratic forms are certain generalizations of the continued fractions algorithm; for example, the algorithm of Voronoi for calculating the units of a cubic field, and the theory of fundamental domains of automorphisms of indefinite quadratic forms.
Let ( rows 1; rows ) and.