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.
FALSE: if and then is nonzero somewhere on
Statement
False claim: the combinatorial Nullstellensatz remains true if the hypothesis that the top monomial coefficient is nonzero is deleted.
Facts & Assumptions
Given: the polynomial and the grid .
The theorem requires the top coefficient to be nonzero (Alon's Combinatorial Nullstellensatz: if , the coefficient of in is nonzero, and , then for some ).
Refutation
The polynomial has total degree , and each grid has size .
But . So the conclusion of the false claim fails.
The missing hypothesis is exactly the coefficient of , which is here. That is why [L1] does not apply.
Depends on
- Alon's Combinatorial Nullstellensatz: if $\deg f=\sum_it_i$, the coefficient of $x_1^{t_1}\cdots x_n^{t_n}$ in $f$ is nonzero, and $\lvert S_i\rvert>t_i$, then $f(s_1,\dots,s_n)\ne0$ for some $s_i\in S_i$
- Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots,x_n]$
- If $\deg_{x_i}P<\lvert S_i\rvert$ for each $i$ and $P$ vanishes on $S_1\times\cdots\times S_n$, then $P=0$
- 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
15 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, Theorem 1.2 (standard reference, not scraped)