Quadratic Invariant -- from Wolfram MathWorld
Author: Weisstein, Eric W.
Description: Given the binary quadratic form ax^2+2bxy+cy^2 (1) with polynomial discriminant b^2-ac, let x = pX+qY (2) y = rX+sY. (3) Then a(pX+qY)^2+2b(pX+qY)(rX+sY)+c(rX+sY)^2 =AX^2+2BXY+CY^2, (4) where A = ap^2+2bpr+cr^2 (5) B = apq+b(ps+qr)+crs (6) C = aq^2+2bqs+cs^2, (7) so B^2-AC=[a^2p^2q^2+b^2(ps+qr)^2+c^2r^2s^2 +2abpq(ps+qr)+2acpqrs+2bcrs(ps+qr)] -(ap^2+2bpr+cr^2)(aq^2+2bqs+cs^2) =a^2p^2q^2+b^2p^2s^2+2b^2pqrs+b^2q^2r^2+c^2r^2s^2 +2abp^2qs+2abpq^2r+2acpqrs+2bcprs^2+2bcqr^2s ...
Keywords: Mathematics:Algebra:Algebraic Invariants

Wolfram Web Resources » 13,454 entries Last updated: Fri Jul 18 2014 Created, developed, and nurtured by Eric Weisstein at Wolfram Research | Algebra > Algebraic Invariants > # Quadratic Invariant Given the binary quadratic form
| (1)
with polynomial discriminant , let
|  |  | (2)
|  |  | (3)
Then
| (4)
where
|  |  | (5)
|  |  | (6)
|  |  | (7)
so
| (8)
Surprisingly, this is the same discriminant as before, but multiplied by the factor . The quantity is called the quadratic invariant modulus. SEE ALSO: Algebraic Invariant, Quadratic CITE THIS AS: Weisstein, Eric W. "Quadratic Invariant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/QuadraticInvariant.html # Wolfram Web Resources
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