Difference between revisions of "Quadratic forms, reduction of"
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− | + | The isolation of "reduced" forms in each class of quadratic forms over a given ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760901.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760903.png" /> are equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760904.png" />, and in the case of their equivalence to find (or describe) all the invertible matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760905.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760906.png" /> taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760907.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760908.png" /> (see [[Quadratic form|Quadratic form]]). For the solution of the latter problem it suffices to know just one such matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q0760909.png" /> and all the automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609010.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609011.png" />, since then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609012.png" />. One usually has in mind equivalence of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609013.png" />, where one is often considering the entire collection of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609014.png" /> and their classes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609015.png" />. There are fundamental differences in the reduction theory of positive-definite and indefinite quadratic forms. | |
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− | 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|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.== | ==The reduction of positive-definite quadratic forms.== | ||
− | There are different methods for the reduction over | + | There are different methods for the reduction over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609017.png" /> and H.F. Charve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609018.png" />. |
− | 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 | To determine a reduced quadratic form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609019.png" /></td> </tr></table> | |
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− | means to define in the positivity cone | + | means to define in the positivity cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609020.png" /> of the coefficient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609022.png" />, a domain of reduction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609024.png" /> is reduced if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609025.png" />. It is desirable that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609026.png" /> possesses good geometric properties (such as simple connectedness, convexity, etc.) and is a fundamental domain of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609027.png" /> of integer transformations of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609028.png" />. A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609029.png" /> is called a fundamental domain of reduction of positive-definite quadratic forms if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609030.png" /> is an open domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609031.png" /> and if: 1) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609032.png" /> there is an equivalent quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609034.png" />) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609035.png" />; and 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609038.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609039.png" />. |
− | of the coefficient space | ||
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− | 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 | ||
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− | for which | ||
− | and 2) if | ||
− | and | ||
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− | then | ||
− | a) Minkowski reduction of a quadratic form. A positive-definite quadratic form | + | a) Minkowski reduction of a quadratic form. A positive-definite quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609040.png" /> is Minkowski reduced if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609041.png" /> and any integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609042.png" /> with greatest common divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609043.png" />, |
− | is Minkowski reduced if for any | ||
− | and any integers | ||
− | with greatest common divisor | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table> | |
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− | From the infinite number of inequalities (1) for the coefficients | + | From the infinite number of inequalities (1) for the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609045.png" /> one can extract a finite number such that the remaining inequalities follow from them. In the coefficient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609046.png" /> 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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609047.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609048.png" /> is a closed set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609049.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609050.png" /> the faces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609051.png" /> have been calculated (see ). |
− | 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) | ||
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− | is a closed set, | ||
− | For | ||
− | the faces of | ||
− | have been calculated (see ). | ||
− | There exists a constant | + | There exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609052.png" /> such that if the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609053.png" /> is Minkowski reduced, then |
− | such that if the quadratic form | ||
− | is Minkowski reduced, then | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609054.png" /></td> </tr></table> | |
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− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609055.png" /> is the determinant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609056.png" />. |
− | is the determinant of | ||
− | Each real positive-definite quadratic form is equivalent over | + | Each real positive-definite quadratic form is equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609057.png" /> 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 [[#References|[8]]], [[#References|[15]]]). |
− | 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 [[#References|[8]]], [[#References|[15]]]). | ||
− | For | + | For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609062.png" />, the conditions of being reduced have the form |
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− | the conditions of being reduced have the form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609063.png" /></td> </tr></table> | |
− | 0 | ||
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− | If one restricts oneself to proper equivalence (when only integer-valued transformations with determinant | + | If one restricts oneself to proper equivalence (when only integer-valued transformations with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609064.png" /> are admitted), then the domain of reduction has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609065.png" /> (the Lagrange–Gauss reduction conditions). The set of all inequivalent (properly-) reduced quadratic forms can be written as the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609066.png" />, where |
− | 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 | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609067.png" /></td> </tr></table> | |
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− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609068.png" /></td> </tr></table> | |
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− | For | + | For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609069.png" /> 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" , |
− | 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" , | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609070.png" /></td> </tr></table> | |
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− | where the integer | + | where the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609071.png" /> is chosen such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609072.png" />. For any real quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609073.png" /> the algorithm is broken up into a finite number of steps. |
− | is chosen such that | ||
− | For any real quadratic form | ||
− | the algorithm is broken up into a finite number of steps. | ||
− | If | + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609075.png" />, with greatest common divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609076.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609077.png" /> there are only two automorphisms (of determinant 1); for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609078.png" />, six automorphisms; and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609079.png" />, four automorphisms. |
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− | 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 | + | b) Venkov reduction of a quadratic form. This is a reduction method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609080.png" />, depending on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609081.png" />, for an arbitrary real positive-definite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609082.png" />-ary quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609083.png" /> (see [[#References|[3]]]). A quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609084.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609086.png" />-reducible if |
− | depending on a parameter | ||
− | for an arbitrary real positive-definite | ||
− | ary quadratic form | ||
− | see [[#References|[3]]]). A quadratic form | ||
− | is said to be | ||
− | reducible if | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609087.png" /></td> </tr></table> | |
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− | for all integer-valued | + | for all integer-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609088.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609089.png" /> of determinant 1; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609090.png" /> is the form reciprocal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609092.png" /> is the quadratic form obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609093.png" /> by the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609094.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609095.png" /> is the Voronoi semi-invariant, defined as follows: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q07609099.png" />, then |
− | matrices | ||
− | of determinant 1; here | ||
− | is the form reciprocal to | ||
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− | is the quadratic form obtained from | ||
− | by the transformation | ||
− | and | ||
− | is the Voronoi semi-invariant, defined as follows: if | ||
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− | then | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090100.png" /></td> </tr></table> | |
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− | |||
− | |||
− | The set of | + | The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090101.png" />-reducible quadratic forms in the coefficient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090102.png" /> is a convex gonohedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090103.png" /> with a finite number of faces lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090104.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090106.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090107.png" /> is the same as the domain of Minkowski reduction. |
− | 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 | + | c) Selling and Charve reduction of a quadratic form. If in the Venkov reduction one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090109.png" /> is the Voronoi first perfect form, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090110.png" /> one obtains the Selling reduction, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090111.png" /> the Charve reduction (see , [[#References|[6]]]). |
− | where | ||
− | is the Voronoi first perfect form, then for | ||
− | one obtains the Selling reduction, and for | ||
− | the Charve reduction (see , [[#References|[6]]]). | ||
==The reduction of indefinite quadratic forms.== | ==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 | + | This is in principle more complicated than that of positive quadratic forms. There are no fundamental domains for them. Only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090112.png" /> is there a definitive reduction theory of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090113.png" />. |
− | is there a definitive reduction theory of quadratic forms over | ||
a) Reduction of indefinite binary quadratic forms. Let | a) Reduction of indefinite binary quadratic forms. Let | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090114.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | be a quadratic form with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090115.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090116.png" /> is not a perfect square. Associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090117.png" /> is the quadratic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090118.png" /> and its distinct irrational roots | |
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090119.png" /></td> </tr></table> | |
− | |||
− | |||
− | The form | + | The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090120.png" /> is said to be reduced if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090123.png" />. These conditions are equivalent to the conditions |
− | is said to be reduced if | ||
− | |||
− | |||
− | These conditions are equivalent to the conditions | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090124.png" /></td> </tr></table> | |
− | 0 | ||
− | |||
− | (and also to the conditions | + | (and also to the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090125.png" />). 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 [[#References|[1]]]). |
− | 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 [[#References|[1]]]). | ||
For a reduced quadratic form there exists precisely one "right neighbouring" and precisely one "left neighbouring" reduced quadratic form (see [[#References|[1]]]). 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. | For a reduced quadratic form there exists precisely one "right neighbouring" and precisely one "left neighbouring" reduced quadratic form (see [[#References|[1]]]). 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 | + | The foregoing theory is valid also for forms with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090126.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090128.png" /> are distinct irrational roots; however, in this case a chain of reduced forms need not be periodic. |
− | 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 | + | All proper automorphisms (of determinant 1) of a quadratic form with greatest common divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090129.png" />, greatest common divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090130.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090131.png" />, have the form |
− | greatest common divisor | ||
− | |||
− | have the form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090132.png" /></td> </tr></table> | |
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090133.png" /></td> </tr></table> | |
− | |||
− | |||
− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090134.png" /> runs through all the solutions of the [[Pell equation|Pell equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090135.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090136.png" /> is the fundamental solution of this equation, that is, the smallest positive solution. Improper automorphisms (of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090137.png" />) exist only for two-sided (or ambiguous) forms, that is, forms whose class coincides with that of its inverse (see [[#References|[1]]]). The subgroup of proper automorphisms of a two-sided form has index 2 in the group of all automorphisms. |
− | runs through all the solutions of the [[Pell equation|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 [[#References|[1]]]). 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 | + | Indefinite integer-valued quadratic forms of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090140.png" />, reduce to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090141.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090142.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090143.png" />. Two quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090144.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090145.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090146.png" />, are properly equivalent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090147.png" />. All the automorphisms of such forms are |
− | |||
− | |||
− | reduce to the form | ||
− | where | ||
− | |||
− | Two quadratic forms | ||
− | and | ||
− | |||
− | are properly equivalent if and only if | ||
− | All the automorphisms of such forms are | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090148.png" /></td> </tr></table> | |
− | |||
(see [[#References|[1]]]). | (see [[#References|[1]]]). | ||
− | b) Reduction of indefinite | + | b) Reduction of indefinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090150.png" />-ary quadratic forms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090151.png" /> be such a form with real coefficients and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090152.png" />. Then there exists a change of variables (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090153.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090154.png" />, such that |
− | ary quadratic forms. Let | ||
− | be such a form with real coefficients and | ||
− | Then there exists a change of variables (over | ||
− | |||
− | such that | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090155.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090156.png" /> is the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090157.png" />. Let |
− | is the signature of | ||
− | Let | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090158.png" /></td> </tr></table> | |
− | |||
− | ( | + | (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090159.png" /> rows 1; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090160.png" /> rows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090161.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090162.png" />. The quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090163.png" /> is associated with the positive-definite quadratic form |
− | rows 1; | ||
− | rows | ||
− | and | ||
− | The quadratic form | ||
− | is associated with the positive-definite quadratic form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090164.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | The form | + | The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090165.png" /> is called (Hermite) reducible if there is a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090166.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090167.png" /> into a sum of squares such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090168.png" /> is (for example, Minkowski) reduced. |
− | 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 [[#References|[13]]], [[#References|[14]]]. Let | + | Equivalent to this definition of a reduced quadratic form is the following [[#References|[13]]], [[#References|[14]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090169.png" /> be the set of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090170.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090171.png" /> of positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090172.png" />-ary quadratic forms satisfying the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090173.png" />. This is a connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090174.png" />-dimensional manifold of the positivity cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090175.png" /> (which can be written out in explicit form). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090176.png" /> be the domain of reduction of positive-definite quadratic forms. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090177.png" /> is called reducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090178.png" /> is non-empty. |
− | 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 | + | The number of classes of integral indefinite quadratic forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090179.png" /> variables with a given determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090180.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090181.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090182.png" /> are equivalent, then there exists an integral transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090183.png" />, the absolute values of the elements of which are bounded by a constant depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090184.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090185.png" />, that takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090186.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090187.png" />. Thus the problem of determining whether or not two indefinite integral quadratic forms are equivalent is solved in a finite number of steps. |
− | 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). | 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 | + | The general form of the automorphisms of a quadratic form was described by Ch. Hermite for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090188.png" /> and by A. Cayley for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090189.png" /> (see [[#References|[10]]]). |
− | and by A. Cayley for arbitrary | ||
− | see [[#References|[10]]]). | ||
− | A fundamental domain has been constructed of the group of automorphisms of an indefinite integral quadratic form | + | A fundamental domain has been constructed of the group of automorphisms of an indefinite integral quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090190.png" /> in a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090191.png" /> bounded by a finite number of algebraic surfaces, and its volume has been calculated [[#References|[13]]]. For the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090192.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090193.png" />-dimensional space a fundamental domain has been constructed of the group of automorphisms of a quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076090/q076090194.png" /> in the form of an infinite pyramid with a finite number of plane faces (see [[#References|[2]]], [[#References|[4]]]). |
− | in a manifold | ||
− | bounded by a finite number of algebraic surfaces, and its volume has been calculated [[#References|[13]]]. 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 [[#References|[2]]], [[#References|[4]]]). | ||
There is a reduction theory of quadratic forms in algebraic number fields (see [[#References|[11]]]). | There is a reduction theory of quadratic forms in algebraic number fields (see [[#References|[11]]]). |
Revision as of 14:53, 7 June 2020
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
,
![]() | (1) |
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 [8], [15]).
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 [3]). 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 , [6]).
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 [1]).
For a reduced quadratic form there exists precisely one "right neighbouring" and precisely one "left neighbouring" reduced quadratic form (see [1]). 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 [1]). 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
![]() |
(see [1]).
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 [13], [14]. 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 [10]).
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 [13]. 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 [2], [4]).
There is a reduction theory of quadratic forms in algebraic number fields (see [11]).
References
[1] | B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) MR0265267 Zbl 0204.37101 |
[2] | B.A. Venkov, Izv. Akad. Nauk SSSR. Ser. Mat. , 1 (1937) pp. 139–170 |
[3] | B.A. Venkov, "The reduction of positive-definite quadratic forms" Izv. Akad. Nauk SSSR. Ser. Mat. , 4 (1940) pp. 37–52 (In Russian) |
[4] | B.A. Venkov, "On indeterminate quadratic forms with integral coefficients" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 30–41 (In Russian) MR0048498 |
[5a] | B.N. Delone, "The geometry of positive definite quadratic forms" Uspekhi Mat. Nauk : 3 (1937) pp. 16–62 (In Russian) |
[5b] | B.N. Delone, "The geometry of positive definite quadratic forms" Uspekhi Mat. Nauk : 4 (1938) pp. 104–164 (In Russian) |
[6] | B.N. Delone, R.V. Galiulin, M.I. Shtorgin, "The types of Bravais lattices" , Current problems in mathematics , 2 , Moscow (1973) pp. 119–254 (In Russian) MR0412947 Zbl 0334.50005 |
[7] | P.G. Lejeune-Dirichlet, "Vorlesungen über Zahlentheorie" , Vieweg (1894) Zbl 25.0252.01 |
[8] | S.S. Ryshkov, "The theory of Hermite–Minkowski reduction of positive definite quadratic forms" J. Soviet Math. , 6 : 6 (1976) pp. 651–671 Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. , 33 (1973) pp. 37–64 Zbl 0374.10019 |
[9a] | P.P. Tammela, "Reduction theory of positive quadratic forms" J. Soviet Math. , 11 : 2 (1979) pp. 197–277 Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. , 50 (1975) pp. 6–96 MR0563103 MR0321875 Zbl 0403.10012 |
[9b] | P.P. Tammela, "Minkowski reduction region for positive quadratic forms in seven variables" J. Soviet Math. , 16 : 1 (1981) pp. 836–857 Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. , 67 (1977) pp. 108–143; 226 Zbl 0453.10033 |
[10] | P. Bachmann, "Zahlentheorie. Die Arithmetik der quadratischen Formen" , 1–2 , Teubner (1923–1925) MR0238661 MR1522322 |
[11] | P. Humbert, "Réduction de formes quadratiques dans un corps algébrique fini" Comm. Math. Helv. , 23 (1949) pp. 50–63 MR0031521 Zbl 0034.31102 |
[12] | H. Minkowski, "Diskontinuitätsbereich für arithmetische Äquivalenz" J. Reine Angew. Math. , 129 (1905) pp. 220–274 Zbl 37.0251.02 |
[13] | C.L. Siegel, "Einheiten quadratischer Formen" Abh. Math. Sem. Univ. Hamburg , 13 (1939) pp. 209–239 MR0003003 Zbl 0023.00701 Zbl 66.0125.03 |
[14] | C.L. Siegel, "Zur Theorie der quadratischen Formen" Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. (1972) pp. 21–46 MR0311578 Zbl 0252.10019 |
[15] | B.L. van der Waerden, "Die Reduktionstheorie der positiven quadratischen Formen" Acta Math. , 96 (1956) pp. 265–309 Zbl 0072.03601 |
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