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Statement

Setting (p. 3). A point $a\in\mathbb F^n$ is a zero of multiplicity $t$ of a nonzero $f\in\mathbb F[X_1,\ldots,X_n]$ when $t$ is the least degree of a term of $f(X_1+a_1,\ldots,X_n+a_n)$; by convention the zero polynomial has a zero of multiplicity $t$ at every point for every positive integer $t$. $T(n,t)$ is the set of nondecreasing sequences $\tau$ of length $t$ on $\{1,\ldots,n\}$, with $i$-th element $\tau(i)$, and $j\in\tau$ means that $j$ appears in $\tau$. The field $\mathbb F$, the sets $S_i$ and the polynomials $g_i$ are as in Theorem 2.1.

Theorem 3.1 (p. 3). If $f$ has a zero of multiplicity $t$ at every point of $S_1\times\cdots\times S_n$, then there are polynomials $h_\tau\in\mathbb F[X_1,\ldots,X_n]$ with $\deg h_\tau\le\deg f-\sum_{i\in\tau}\deg g_i$ such that

$$f=\sum_{\tau\in T(n,t)}g_{\tau(1)}\cdots g_{\tau(t)}h_\tau.$$

Reading. Two points of the print are read as the paper uses them. The sum $\sum_{i\in\tau}\deg g_i$ counts a repeated index as often as it occurs in $\tau$, that is $\sum_{k=1}^t\deg g_{\tau(k)}$: the proof of Corollary 3.2 (p. 4) counts the occurrences of each index in $\tau$. Multiplicity $t$ is read as multiplicity at least $t$: the induction on p. 4 applies the theorem to a quotient whose multiplicity it bounds only from below.

Proof pointer

Pp. 3–4, a double induction on $n$ and $t$, after Bruen's proof of his Theorem 1.3. The base cases are $n=1$, where $g_1^t$ divides $f$, and $t=1$, which is Theorem 2.1. Dividing successively by $X_n-\alpha$ for the elements $\alpha$ of $S_n$, and applying the case of $n-1$ variables to each remainder, gives $f=g_n(X_n)A+B$, where $B$ has the required form over $T(n-1,t)$ and $\deg A\le\deg f-\deg g_n$. Then $A$ has a zero of multiplicity $t-1$ on the grid, and the case $t-1$ applies to it.

Read depth

Claims checked: the definitions and the statement were read clause by clause against pp. 3–4 of the print, and the proof was followed.

Dependencies

Theorem 2.1. The paper bases its proof on A. A. Bruen, Polynomial multiplicities over finite fields and intersection sets, J. Combin. Theory Ser. A 60 (1992), 19–33.

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.