Reduction of Ternary and Quartic Quadratic Forms

Binary Quadratic Forms: A integer quadratic form in two variables f(X,Y)=aX^2+bXY+cY^2 is (properly) equivalent to another form g(X,Y) is one is obtained from the other by integer change of coordinates (with determinant 1). We can to find some “small” representatives in each of the equivalence classes. So we uses some equivalences(change of coordinates) to reduce a given form to a simpler form.

a) If f is a positive definite form with D=b^2-4ac <0, a>0 , it’s called reduced iff

\displaystyle |b| \le a \le c and in addition we have \displaystyle a=c \text{ or } |b|=a \implies b \ge 0 .

One can reduce using the moves:


i) Shift: \displaystyle [a, b, c] \to [a, b', c'], b' = b \mod 2a, |b'| \le a and
ii) Invert: \displaystyle [a, b, c] \to [c, -b, a]

Each proper equivalence class is represented by a unique reduced form.

Example: \displaystyle [6, -5, 4] \to [4, 5, 6] \to [4, -3, 5]

\displaystyle [4, -3, 5] ~~ is the reduced forms of discriminant -71 equivalent to the given form 6x^2-5xy+4y^2. All the reduced form are given by

\displaystyle x^2 + xy + 18y^2, 2x^2 - xy + 9y^2, 2x^2 + xy + 9y^2, 3x^2 - xy + 6y^2, 3x^2 + xy + 6y^2, 4x^2 - 3xy + 5y^2, 4x^2 + 3xy + 5y^2

Let say D=-20, then there are only two reduced forms [1, 0, 5], [2, 2, 3] which represent the 2 proper equivalence classes.

By the map \displaystyle [a, b, c] \to \frac{-b + \sqrt{D}}{2a} , the we represent positive definite forms with a point in the upper half-plane \displaystyle \mathbb H =\{z=x+iy:y>0\} , and the above reduction algorithm is the reduction of these points into the Fundamental domain \mathbb H/SL_2(\mathbb Z)


b) If f is an indefinite forms (D=b^2-4ac >0 ) where D is not a square , it’s reduced iff

\displaystyle |2| a|-\sqrt{d}|<b<\sqrt{D}
Equivalently the roots of f(X,1) satisfy |r_1| < 1<|r_2|, r_1r_2 <0

We can reduce by moving to adjacent form \displaystyle [a, b, c] \to [c, b', c'] , b'=-b \mod 2c where

i) \displaystyle -|c|<b^{\prime} \leq|c| if \displaystyle |c|>\sqrt{d}
ii) \displaystyle \sqrt{d}-2|c|<b^{\prime}<\sqrt{d} if \displaystyle |c|<\sqrt{d} .

Every equivalence class is represented by a cycle of adjacent reduced forms (not by a unique form).

Example: D=12 has four reduced forms [-2,2,1],[1,2,-2],[2,2,-1], [-1,2,2] but we have the cycles

\displaystyle [-2, 2, 1] \to [1, 2, -2] \to [-2, 2, 1] \to [1, 2, -2] \to [-2, 2, 1] \cdots
\displaystyle [2, 2, -1] \to [-1, 2, 2] \to [2, 2, -1] \to [-1, 2, 2] \to [2, 2, -1] \to [-1, 2, 2] \cdots

which represent the 2 proper equivalence classes.

In the indefinite case, both the roots are real and not in the upper half-plane. In this case, we take the semi-cirlce on the upper-half plane joining the roots (geodesic) which will represent the quadratic form.

We can similarly define equivalence of quadratic forms in many variables.

Ternary Quadratic Form:

Positive definite forms: That is forms with

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