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Statement

Setting (p. 5). Let $\mathbb F$ be a field and $f\in\mathbb F[X_1,\ldots,X_n]$. For $i=1,\ldots,n$ let $D_i$ and $S_i$ be finite nonempty subsets of $\mathbb F$ with $D_i\subset S_i$, and put

$$g_i(X_i)=\prod_{s\in S_i}(X_i-s),\qquad l_i(X_i)=\prod_{d\in D_i}(X_i-d).$$

Multiplicity, $T(n,t)$ and the sums over $i\in\tau$ are as on the Theorem 3.1 page.

Theorem 4.1 (p. 5). Suppose $f$ has a zero of multiplicity at least $t$ at every point of $S_1\times\cdots\times S_n$ except at at least one point of $D_1\times\cdots\times D_n$, where its multiplicity is less than $t$. Then there are polynomials $h_\tau\in\mathbb F[X_1,\ldots,X_n]$ with $\deg h_\tau\le\deg f-\sum_{i\in\tau}\deg g_i$, and a nonzero polynomial $u$ with $\deg u\le\deg f-\sum_{i=1}^n(\deg g_i-\deg l_i)$, such that

$$f=\sum_{\tau\in T(n,t)}g_{\tau(1)}\cdots g_{\tau(t)}h_\tau+u\prod_{i=1}^n\frac{g_i}{l_i}.$$

If moreover $f$ is nonzero at some point of $D_1\times\cdots\times D_n$, then

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

Reading. The print writes $\deg(h_i)$ for the coefficients $h_\tau$. The hypothesis is read as multiplicity at least $t$ at every point of $(S_1\times\cdots\times S_n)\setminus(D_1\times\cdots\times D_n)$ and multiplicity less than $t$ at some point of $D_1\times\cdots\times D_n$; the proof uses it in this form. For $t>1$ the degree bound needs the stronger hypothesis that $f$ is nonzero at a point of $D_1\times\cdots\times D_n$.

Proof pointer

Pp. 5–6. Reduce $f$ modulo the ideal generated by the products $g_{\tau(1)}\cdots g_{\tau(t)}$ to a remainder $w$, which is nonzero because $f$ is not in that ideal. For each $i$, $fl_i^t$ and hence $wl_i^t$ have a zero of multiplicity $t$ on the whole grid, and Theorem 3.1 together with the form of $w$ shows that $g_i$ divides $wl_i^t$; since $g_i/l_i$ is coprime to $l_i$, it divides $w$. For the degree bound the paper fixes an index $j$ attaining the maximum and a point $d$ of the smaller grid where $f$ is nonzero, and shows that $(g_j/l_j)^{t-1}$ divides the nonzero one-variable restriction of $u$ through $d$.

Read depth

Claims checked: the statement was read clause by clause against pp. 5–6 of the print, and the proof was followed.

Dependencies

Theorem 3.1. The 2011 erratum restates this theorem and corrects only Corollary 4.2.

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.