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Quadratic Invariant

Given the Binary Quadratic Form

\begin{displaymath}
ax^2+2bxy+cy^2
\end{displaymath} (1)

with Discriminant $b^2-ac$, let
$\displaystyle x$ $\textstyle =$ $\displaystyle pX+qY$ (2)
$\displaystyle y$ $\textstyle =$ $\displaystyle rX+sY.$ (3)

Then


\begin{displaymath}
a(pX+qY)^2+2b(pX+qY)(rX+sY)+c(rX+sY)^2=AX^2+2BXY+CY^2,
\end{displaymath} (4)

where
$\displaystyle A$ $\textstyle =$ $\displaystyle ap^2+2bpr+cr^2$ (5)
$\displaystyle B$ $\textstyle =$ $\displaystyle apq+b(ps+qr)+crs$ (6)
$\displaystyle C$ $\textstyle =$ $\displaystyle aq^2+2bqs+cs^2,$ (7)

so
$B^2-AC=[a^2p^2q^2+b^2(ps+qr)^2+c^2r^2s^2$
$\quad\phantom{=} +2abpq(ps+qr)+2acpqrs+2bcrs(ps+qr)]$
$\quad\phantom{=} -(ap^2+2bpr+cr^2)(aq^2+2bqs+cs^2)$
$\quad = a^2p^2q^2+b^2p^2s^2+2b^2pqrs+b^2q^2r^2+c^2r^2s^2$
$\quad\phantom{=} +2abp^2qs+2abpq^2r+2acpqrs+2bcprs^2+2bcqr^2s$
$\quad\phantom{=} -a^2p^2q^2-2abp^2qs-acp^2s^2-2abpq^2r-4b^2pqrs$
$\quad\phantom{=} -2bcprs^2-acq^2r^2-2bcqr^2s-c^2r^2s^2$
$\quad = b^2p^2s^2-2b^2pqrs+b^2q^2r^2+2acpqrs-acp^2s^2$
$\quad\phantom{=} -acq^2r^2$
$\quad = p^2s^2(b^2-ac)+q^2r^2(b^2-ac)-2pqrs(b^2-ac)$
$\quad = (b^2-ac)(p^2s^2-2pqrs+q^2r^2)$
$\quad = (ps-rq)^2(b^2-ac).$ (8)
Surprisingly, this is the same discriminant as before, but multiplied by the factor $(ps-rq)^2$. The quantity $ps-rq$ is called the Modulus.

See also Algebraic Invariant




© 1996-9 Eric W. Weisstein
1999-05-25