Examples of Reducing Binary Quadratic Forms 2

Recall from the Reduced Binary Quadratic Forms page that a positive definite binary quadratic form $f(x, y) = ax^2 + bxy + cy^2$ is said to be reduced if either:

• 1) $-a < b \leq a < c$.

• 2) $0 < b < a = c$.

Also recall that $M_1 = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$ and $M_2 = M_2 = \begin{bmatrix} 1 & m\\ 0 & 1 \end{bmatrix}$ and if $a > c$ then applying $M_1$ to $f(x, y)$ gives us an equivalent binary quadratic form with $a < c$. If $b > a$ (equivalently $-a > b$) then applying $M_2$ to $f(x, y)$ gives us an equivalent binary quadratic form with $b < a$ (equivalently $-a < b$). We will now look at some examples of taking a binary quadratic form and finding an equivalent binary quadratic form that is reduced.

Example 1

Reduce the binary quadratic form $g(x, y) = 5x^2 - 6xy +6y^2$.

We have that $a = 5$, $b = -6$, and $c = 6$. So:

(1)

\begin{align} \quad -5 \not < -6 \leq 5 < 6 \end{align}

So we want to find the integer $m$ such that:

(2)

\begin{align} \quad -5 < 2(5)m - 6 < 5 \quad \Leftrightarrow \quad -5 < 10m - 6 < 5 \end{align}

So $m = 1$. Therefore using the matrix $M_2 = \begin{bmatrix} 1 & m \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ to get:

(3)

\begin{align} \quad f(x, y) = g(x + y, y) = 5(x + y)^2 - 6(x + y)y +6y^2 = 5x^2 + 10xy + 5y^2 - 6xy - 6y^2 + 6y^2 = 5x^2 + 4xy + 5y^2 \end{align}

Observe that $f(x, y)$ is such that $0 < 4 < 5 = 5$, and so $f(x, y)$ is equivalent to the reduced binary quadratic form $g(x, y) = 5x^2 + 4xy + 5y^2$.

Example 2

Reduce the binary quadratic form $h(x, y) = 11x^2 - 2xy + 2y^2$.

We have that $a = 11$, $b = -2$, and $c = 2$. So $-11 < -2 \leq 11 \not < 2$.

We apply the matrix $M_1 = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$ to get:

(4)

\begin{align} \quad g(x, y) = h(-y, x) = 11(-y)^2 - 2(-y)(x) + 2(x)^2 = 2x^2 + 2xy + 11y^2 \end{align}

Observe that $g(x, y)$ is such that $-2 < 2 \leq 2 < 11$. Therefore $g(x, y)$ is a reduced binary quadratic form that is equivalent to $h(x, y)$.