Alphabeta MathHow 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.
Formal polynomials are not the functions they induce
A formal polynomial is its finitely supported coefficient sequence (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution). Evaluation (Evaluation and roots of a polynomial in a commutative target ring) assigns to it a function only after a target ring and a coefficient homomorphism have been chosen. Distinct formal polynomials can therefore induce the same function on a finite ring.
Over an infinite integral domain the distinction remains conceptual but evaluation is injective: if two polynomials have equal values, their difference has every domain element as a root, and A nonzero polynomial of degree over an integral domain has at most distinct roots forces that difference to be zero. The infinitude and domain hypotheses are both essential to that conclusion.
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Used by
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Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 27 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Neil Donaldson, Math 120B Notes, Section 22, Golden Rule (standard reference, not scraped)