AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exact dyadic determinant sign by integer clearance

Definition

Input: a finite $s \times s$ matrix $M$ of exact dyadic rationals with $s \geq 1$ . Output: the sign of its exact determinant, including zero.

In row $i$ , write each nonzero entry in reduced form $M_{i j} = a_{i j} / 2^{b_{i j}}$ in reduced form with $b_{i j} \geq 0$ and $a_{i j}$ odd when $b_{i j} > 0$ ; write a zero entry with $b_{i j} = 0$ . Set $k_{i} = \max_{j} b_{i j}$ and form $N_{i j} = 2^{k_{i}} M_{i j}$ , an integer matrix.
Compute $\sum_{π \in S_{s}} sgn (π) \prod_{i} N_{i, π (i)}$ using arbitrary-precision integer arithmetic.
Return the sign of this integer sum.

Depends on

Used by

Dependency tree · two levels

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Sources

Jonathan R. Shewchuk, Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates, §4 (standard reference, not scraped)
CGAL Developer Manual, Robustness Issues, §11.1 (standard reference, not scraped)