Binary Quadratic Form
A binary quadratic form is a quadratic form in two variables having the form
| Q(x,y)=ax^2+2bxy+cy^2, |
(1)
|
commonly denoted <a,b,c>.
Consider a binary quadratic form with real coefficients a, b, and c, determinant
| D=b^2-ac=1, |
(2)
|
and a>0. Then Q(x,y) is positive definite. An important result states that there exist two integers x and y not both 0 such that
| [画像: Q(x,y)<=2/(sqrt(3)) ] |
(3)
|
for all values of a, b, and c satisfying the above constraint (Hilbert and Cohn-Vossen 1999, p. 39).
See also
Binary Quadratic Form Determinant, Binary Quadratic Form Discriminant, Pell Equation, Positive Definite Quadratic Form, Quadratic Form, Quadratic InvariantExplore with Wolfram|Alpha
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References
Hilbert, D. and Cohn-Vossen, S. "The Minimum Value of Quadratic Forms." §6.2 in Geometry and the Imagination. New York: Chelsea, pp. 39-41, 1999.Referenced on Wolfram|Alpha
Binary Quadratic FormCite this as:
Weisstein, Eric W. "Binary Quadratic Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BinaryQuadraticForm.html