Quadratic forms, reduction of
The isolation of "reduced" forms in each class of quadratic forms over a given ring , i.e. of (one or several) "standard" forms in the class. The main aim of the reduction of quadratic forms is the solution of the problem of equivalence of quadratic forms: To establish whether or not two given quadratic forms and are equivalent over , and in the case of their equivalence to find (or describe) all the invertible matrices over taking to (see Quadratic form). For the solution of the latter problem it suffices to know just one such matrix and all the automorphisms of the form , since then . One usually has in mind equivalence of quadratic forms over , where one is often considering the entire collection of quadratic forms over and their classes over . There are fundamental differences in the reduction theory of positive-definite and indefinite quadratic forms.
The reduction of positive-definite quadratic forms.
There are different methods for the reduction over of real positive-definite quadratic forms. Of these the most extensive and widely studied is the Minkowski (or Hermite–Minkowski) reduction method. The most general method is Venkov's method. Other prevalent reductions are those of E. Selling and H.F. Charve .
To determine a reduced quadratic form
means to define in the positivity cone of the coefficient space , , a domain of reduction such that is reduced if and only if . It is desirable that possesses good geometric properties (such as simple connectedness, convexity, etc.) and is a fundamental domain of the group of integer transformations of determinant . A domain is called a fundamental domain of reduction of positive-definite quadratic forms if is an open domain in and if: 1) for each there is an equivalent quadratic form () for which ; and 2) if and (), then .
a) Minkowski reduction of a quadratic form. A positive-definite quadratic form is Minkowski reduced if for any and any integers with greatest common divisor ,
From the infinite number of inequalities (1) for the coefficients one can extract a finite number such that the remaining inequalities follow from them. In the coefficient space the set of Minkowski-reduced forms is an infinite complex pyramid (a gonohedron) with a finite number of faces, called the domain of Minkowski reduction (or Hermite–Minkowski gonohedron) ; is a closed set, . For the faces of have been calculated (see ).
There exists a constant such that if the quadratic form is Minkowski reduced, then
where is the determinant of .
Each real positive-definite quadratic form is equivalent over to a Minkowski-reduced quadratic form. There is an algorithm for the reduction (for finding a reduced form that is equivalent to a given one) (see , ).
For , , , , , the conditions of being reduced have the form
If one restricts oneself to proper equivalence (when only integer-valued transformations with determinant are admitted), then the domain of reduction has the form (the Lagrange–Gauss reduction conditions). The set of all inequivalent (properly-) reduced quadratic forms can be written as the union , where
For there is an algorithm for Gauss reduction, according to which one has to go over from a form not satisfying the Lagrange–Gauss conditions to its "neighbour" ,
where the integer is chosen such that . For any real quadratic form the algorithm is broken up into a finite number of steps.
If , , with greatest common divisor , then for there are only two automorphisms (of determinant 1); for , six automorphisms; and for , four automorphisms.
b) Venkov reduction of a quadratic form. This is a reduction method , depending on a parameter , for an arbitrary real positive-definite -ary quadratic form (see ). A quadratic form is said to be -reducible if
for all integer-valued -matrices of determinant 1; here is the form reciprocal to , is the quadratic form obtained from by the transformation , and is the Voronoi semi-invariant, defined as follows: if , , , , then
The set of -reducible quadratic forms in the coefficient space is a convex gonohedron with a finite number of faces lying in . If and , then is the same as the domain of Minkowski reduction.
c) Selling and Charve reduction of a quadratic form. If in the Venkov reduction one puts , where is the Voronoi first perfect form, then for one obtains the Selling reduction, and for the Charve reduction (see , ).
The reduction of indefinite quadratic forms.
This is in principle more complicated than that of positive quadratic forms. There are no fundamental domains for them. Only for is there a definitive reduction theory of quadratic forms over .
a) Reduction of indefinite binary quadratic forms. Let
be a quadratic form with determinant , where is not a perfect square. Associated with is the quadratic equation and its distinct irrational roots
The form is said to be reduced if , , . These conditions are equivalent to the conditions
(and also to the conditions ). The number of reduced integer-valued quadratic forms of given determinant is finite. Every quadratic form is equivalent to a reduced one. There is an algorithm for reduction, using continued fractions (see ).
For a reduced quadratic form there exists precisely one "right neighbouring" and precisely one "left neighbouring" reduced quadratic form (see ). By going over from a reduced quadratic form to its "neighbouring" , one obtains a doubly-infinite chain of reduced forms. This chain is periodic. A finite segment of inequivalent forms of this chain is called a period. Two reduced forms are properly equivalent if and only if one of them is in the period of the other.
The foregoing theory is valid also for forms with real coefficients if and are distinct irrational roots; however, in this case a chain of reduced forms need not be periodic.
All proper automorphisms (of determinant 1) of a quadratic form with greatest common divisor , greatest common divisor , , have the form
where runs through all the solutions of the Pell equation and is the fundamental solution of this equation, that is, the smallest positive solution. Improper automorphisms (of determinant ) exist only for two-sided (or ambiguous) forms, that is, forms whose class coincides with that of its inverse (see ). The subgroup of proper automorphisms of a two-sided form has index 2 in the group of all automorphisms.
Indefinite integer-valued quadratic forms of determinant , , , reduce to the form , where , . Two quadratic forms and , , are properly equivalent if and only if . All the automorphisms of such forms are
b) Reduction of indefinite -ary quadratic forms. Let be such a form with real coefficients and . Then there exists a change of variables (over ), , such that
where is the signature of . Let
( rows 1; rows ) and . The quadratic form is associated with the positive-definite quadratic form
The form is called (Hermite) reducible if there is a transformation of the form into a sum of squares such that is (for example, Minkowski) reduced.
Equivalent to this definition of a reduced quadratic form is the following , . Let be the set of matrices over of positive -ary quadratic forms satisfying the equation . This is a connected -dimensional manifold of the positivity cone (which can be written out in explicit form). Let be the domain of reduction of positive-definite quadratic forms. The form is called reducible if is non-empty.
The number of classes of integral indefinite quadratic forms in variables with a given determinant is finite (this is true also for positive-definite quadratic forms). The number of reduced forms in a given class is also finite. 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 to . Thus the problem of determining whether or not two indefinite integral quadratic forms are equivalent is solved in a finite number of steps.
c) Automorphisms of indefinite quadratic forms. The problem of the description of all automorphisms of an indefinite integral quadratic form has two aspects: 1) to construct a fundamental domain of the group of automorphisms; 2) to describe the general form of the automorphisms (similar to the description of automorphisms by means of the Pell equation).
The general form of the automorphisms of a quadratic form was described by Ch. Hermite for and by A. Cayley for arbitrary (see ).
A fundamental domain has been constructed of the group of automorphisms of an indefinite integral quadratic form in a manifold bounded by a finite number of algebraic surfaces, and its volume has been calculated . For the case in the -dimensional space a fundamental domain has been constructed of the group of automorphisms of a quadratic form in the form of an infinite pyramid with a finite number of plane faces (see , ).
There is a reduction theory of quadratic forms in algebraic number fields (see ).
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