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Statement

Theorem 5.3 (p. 8). Let $A$ be a set of hyperplanes of $\operatorname{AG}(n,\mathbb F)$, and for $1\le i\le n$ let $D_i$ be a nonempty proper subset of a finite set $S_i\subseteq\mathbb F$. Suppose every point $(s_1,\ldots,s_n)$ with $s_i\in S_i$ lies on at least $t$ hyperplanes of $A$, except at least one point of $D_1\times\cdots\times D_n$, which lies on no hyperplane of $A$. Then

$$|A|\ge(t-1)\max_j\bigl(|S_j|-|D_j|\bigr)+\sum_{i=1}^n\bigl(|S_i|-|D_i|\bigr).$$

The hypothesis is read as for Theorem 4.1: points outside $D_1\times\cdots\times D_n$ lie on at least $t$ hyperplanes of $A$, and at least one point of $D_1\times\cdots\times D_n$ lies on none. The paper calls the theorem almost the dual of Theorem 5.1.

The remark after the theorem (p. 8). With $S_i=\mathbb F_q$ and $D_i=\{0\}$ the paper states that a set of hyperplanes covering every point of $\operatorname{AG}(n,q)$ other than the origin at least $t$ times has at least $(n+t-1)(q-1)$ members, and calls this the dual of Theorem 5.2. The theorem gives this only when the origin lies on no hyperplane of the set, a condition the remark leaves out and cannot drop: for $n=2$, $t=1$ and $q>2$, the $q$ lines $X_1=a$ cover the whole plane and $q<2(q-1)$.

Proof pointer

P. 8. The product $f$ of the affine linear forms defining the hyperplanes of $A$ has degree $|A|$, a zero of multiplicity at least $t$ at the covered points, and a nonzero value at the uncovered point, so Theorem 4.1 applies.

Read depth

Claims checked: the statement and the remark after it were read clause by clause against p. 8 of the print, and the proof was followed.

Dependencies

Theorem 4.1.

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.