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Reduced positive-definite binary quadratic forms
Definition
A positive-definite binary quadratic form (Positive-definite binary quadratic forms) is reduced when
and, in the two boundary cases, the middle coefficient is required to be nonnegative:
Remarks
- The inequalities alone are not enough for uniqueness: the sign convention at the boundary removes the duplicate reduced representatives.
- The definition applies only to positive-definite forms; the companion page records that the indefinite theory uses a different convention and does not have uniqueness.
Depends on
Used by
- Distinct reduced forms can represent the same integers Counterexample
- Reducing (458,214,25) to (1,0,1) Example
- A non-reduced positive-definite form admits an equivalent positive-definite form with smaller reduction measure Lemma
- A reduced positive-definite form of discriminant Δ satisfies a≤√|Δ|/3 Lemma
- Properly equivalent reduced forms with the same leading coefficient are equal Lemma
- The leading coefficient of a reduced positive-definite form is minimal in its proper-equivalence class Lemma
- Every positive-definite integral binary quadratic form is properly equivalent to a reduced form Theorem
Dependency tree · two levels
2 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, Definition 9.3.1 (standard reference, not scraped)
- Andrew Granville, Binary Quadratic Forms, Chapter 4 (standard reference, not scraped)