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An integral binary quadratic form is positive definite exactly when its leading coefficient is positive and its discriminant is negative

Statement

Let $f (x, y) = a x^{2} + b x y + c y^{2}$ be an integral binary quadratic form, and let $Δ = b^{2} - 4 a c$ be its discriminant. Then $f$ is positive definite if and only if

$a > 0 and Δ < 0.$

Facts & Assumptions

Given: The integral binary quadratic form $f (x, y) = a x^{2} + b x y + c y^{2}$ and its discriminant $Δ = b^{2} - 4 a c$ .

[F1]

A form is positive definite when $f (x, y) > 0$ for every real pair $(x, y) \neq (0, 0)$ (Positive-definite binary quadratic forms).

[F2]

The discriminant is $Δ = b^{2} - 4 a c$ (The discriminant of a binary quadratic form).

Proof

technique · direct

1.1F1

Suppose $f$ is positive definite. Then $a = f (1, 0) > 0$ by [F1].

1.2F2givenalgebra

Conversely, suppose $a > 0$ and $Δ < 0$ . For every real $(x, y)$ one has $4 a f (x, y) = (2 a x + b y)^{2} - Δ y^{2}$ by direct expansion.

1.3givenalgebra

If $(x, y) \neq (0, 0)$ and $y = 0$ , then $x \neq 0$ and $f (x, 0) = a x^{2} > 0$ because $a > 0$ .

2.1F1F2step 1.1algebra

Also $(- \frac{b}{2 a}, 1) \neq (0, 0)$ , so [F1] gives $f (- \frac{b}{2 a}, 1) > 0$ . Multiplying by $4 a > 0$ from step 1.1 yields $- Δ = (2 a (- b / (2 a)) + b)^{2} - Δ > 0$ , hence $Δ < 0$ .

2.2step 1.2givenalgebra

If $(x, y) \neq (0, 0)$ and $y \neq 0$ , then step 1.2 gives $4 a f (x, y) = (2 a x + b y)^{2} + (- Δ) y^{2} > 0$ because both summands are nonnegative and the second is positive.

3.1F1step 1.1step 2.1step 2.2step 1.3∎

Steps 2.2 and 1.3 cover every nonzero real pair, so $f$ is positive definite by [F1]. Together with steps 1.1 and 2.1, this proves the criterion.

Depends on

Used by

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3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

William Stein, Elementary Number Theory and Elliptic Curves, Section 9.2.4 (standard reference, not scraped)
Andrew Granville, Binary Quadratic Forms, Chapter 4 (standard reference, not scraped)