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Statement

Setting (p. 2). Let $\mathbb F$ be a field and $f$ a nonzero polynomial in $\mathbb F[X_1,\ldots,X_n]$. Let $S_1,\ldots,S_n$ be arbitrary nonempty finite subsets of $\mathbb F$, and put $g_i(X_i)=\prod_{s\in S_i}(X_i-s)$, so that $\deg g_i=|S_i|$.

Theorem 2.1 (p. 2). If $f(s_1,\ldots,s_n)=0$ for every choice of $s_i\in S_i$, then there are polynomials $h_1,\ldots,h_n\in\mathbb F[X_1,\ldots,X_n]$ with $\deg h_i\le\deg f-\deg g_i$ such that

$$f=\sum_{i=1}^n h_ig_i.$$

The paper quotes this as Alon's Combinatorial Nullstellensatz, citing N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999), 7–29, Theorem 1.1, and gives no proof of it.

Proof pointer

None in the paper; the result is an external input.

Read depth

Claims checked: the statement was read clause by clause against p. 2 of the print.

Dependencies

None in the corpus. External input: Alon (1999), Theorem 1.1.

Source. Simeon Ball and Oriol Serra, Punctured combinatorial Nullstellensätze, Combinatorica 29 (2009), 511–522, doi:10.1007/s00493-009-2509-z. Labels and page numbers are those of the corrected author manuscript dated 14 June 2011, the edition named on the source card.

Bears on

None recorded. The paper names no Erdős problem.