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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

 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 Invariant

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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.

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Binary Quadratic Form

Cite this as:

Weisstein, Eric W. "Binary Quadratic Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinaryQuadraticForm.html

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