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.
vanishes on although
Statement refuted
The strict inequality in the polynomial identity lemma cannot be weakened to equality.
Facts & Assumptions
Given: a field , the polynomial , and the set .
Over a field, if a polynomial has degree in each variable strictly below the size of the corresponding finite grid set and vanishes on the whole grid, then it is the zero polynomial (If for each and vanishes on , then ).
Counterexample
The polynomial is nonzero and has degree .
Yet and , so vanishes on all of .
Therefore the conclusion of [L1] fails when the strict inequality is replaced by equality.
Depends on
- If $\deg_{x_i}P<\lvert S_i\rvert$ for each $i$ and $P$ vanishes on $S_1\times\cdots\times S_n$, then $P=0$
- Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots,x_n]$
- A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots
- Evaluation and roots of a polynomial in a commutative target ring
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
17 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
- N. Alon, Combinatorial Nullstellensatz, Lemma 2.1 (standard reference, not scraped)