Let $p$ be a prime and $b_1, \ldots, b_p \in \mathbb{Z}_p$ with $\sum_{i=1}^p b_i = 0$. Then there exist enumerations $a_1, \ldots, a_p$ and $c_1, \ldots, c_p$ of $\mathbb{Z}_p$ such that $a_i + b_i = c_i$ for each $i$.

If $p = 2n + 1$ is prime, then for any prescribed distances $d_1, \ldots, d_n \in \{1, \ldots, n\}$, the $n$ couples can be seated at the $2n$ non-host positions of $\mathbb{Z}_p$ such that couple $i$ occupies positions $\{x_i, x_i + d_i\}$, with the $2n$ occupied positions all distinct.

Let $p$ be a prime and $A \subseteq \mathbb{Z}_p$ non-empty with $2|A| \leq p + 3$. Then \begin{align*} |A \mathbin{\dot{+}} A| \geq 2|A| - 3. \end{align*}

Let $p$ be a prime and $A, B \subseteq \mathbb{Z}_p$ with $2 \leq |A| < |B|$ and $|A| + |B| \leq p + 2$. Then \begin{align*} |A \mathbin{\dot{+}} B| \geq |A| + |B| - 2. \end{align*}

Let $f \in \mathbb{F}_q[X_1, \ldots, X_n]$ have degree $< n$. Then the number of zeros $N(f)$ of $f$ is divisible by $p$.

Let $f_1, \ldots, f_m \in \mathbb{F}_q[X_1, \ldots, X_n]$ satisfy $\sum_{i=1}^m \deg f_i < n$. Then the polynomials $f_1, \ldots, f_m$ cannot have exactly one common zero.

Let $\mathbb{F}$ be a field, and let $S_1, \ldots, S_n$ be non-empty finite subsets of $\mathbb{F}$ with $|S_i| = d_i + 1$. Let $f \in \mathbb{F}[X_1, \ldots, X_n]$ have degree $d = \sum_{i=1}^n d_i$, and suppose the coefficient of $X_1^{d_1} \cdots X_n^{d_n}$ in $f$ is nonzero. Then $f$ is not identically zero on $S = S_1 \times \cdots \times S_n$.

For $i = 1, \ldots, n$, let $S_i$ be a non-empty finite subset of $\mathbb{F}$, and set \begin{align*} g_i(X_i) = \prod_{s \in S_i}(X_i - s) \in \mathbb{F}[X_i] \subseteq \mathbb{F}[X_1, \ldots, X_n]. \end{align*} If $f \in \mathbb{F}[X_1, \ldots, X_n]$ vanishes identically on $S = S_1 \times \cdots \times S_n$, then there exist $h_i \in \mathbb{F}[X_1, \ldots, X_n]$ with $\deg h_i \leq \deg f - |S_i|$ such that \begin{align*} f = \sum_{i=1}^{n} h_i g_i. \end{align*}

Let $S_1, \ldots, S_n$ be non-empty finite subsets of a field $\mathbb{F}$, and let $h \in \mathbb{F}[X_1, \ldots, X_n]$ satisfy $\deg_{X_i} h < |S_i|$ for each $i = 1, \ldots, n$. If $h$ vanishes identically on $S = S_1 \times \cdots \times S_n$, then $h$ is the zero polynomial.

Let $f \in R[X_1, \ldots, X_n]$, and for $i = 1, \ldots, n$ let $g_i(X_i) \in R[X_i]$ be monic of degree $d_i$. Then there exist $h_1, \ldots, h_n, r \in R[X_1, \ldots, X_n]$ such that \begin{align*} f = \sum_{i=1}^{n} h_i g_i + r, \end{align*} where for all $i, j$: \begin{align*} \deg h_i &\leq \deg f - d_i, \qquad \deg_{X_i} h_i \leq \deg_{X_i} f - d_i, \\ \deg_{X_j} h_i &\leq \deg_{X_j} f, \qquad \deg_{X_i} r \leq d_i - 1, \\ \deg r &\leq \deg f. \end{align*}