[proofplan] We encode the seating conditions as the nonvanishing of a polynomial $f \in \mathbb{Z}_p[X_1, \ldots, X_n]$ that is a product of linear factors: factors $X_i$ and $X_i + d_i$ ensure couple $i$ avoids the host's seat, and factors $(X_i - X_j)$, $(X_i + d_i - X_j)$, $(X_i - X_j - d_j)$, $(X_i + d_i - X_j - d_j)$ for $i < j$ ensure all $2n$ occupied positions are distinct. The total degree is $2n^2 = n(p-1)$, matching $\sum d_i$ for $d_i = p - 1 = 2n$ and $S_i = \mathbb{Z}_p$. The coefficient of $X_1^{2n} \cdots X_n^{2n}$ reduces to a multinomial coefficient that is nonzero modulo $p$ by Wilson's theorem, so [Alon's Combinatorial Nullstellensatz](/theorems/2617) guarantees a nonzero of $f$. [/proofplan]

[step:Build the polynomial whose nonzeros are valid seatings]We work in $\mathbb{Z}_p$ where $p = 2n + 1$. A valid seating assigns starting positions $x_1, \ldots, x_n \in \mathbb{Z}_p$ such that the $2n$ values $x_1, x_1 + d_1, x_2, x_2 + d_2, \ldots, x_n, x_n + d_n$ are all distinct and all nonzero (the host occupies position $0$). Define \begin{align*} f(X_1, \ldots, X_n) = \prod_{i=1}^n X_i(X_i + d_i) \cdot \prod_{1 \leq i < j \leq n} (X_i - X_j)(X_i + d_i - X_j)(X_i - X_j - d_j)(X_i + d_i - X_j - d_j). \end{align*} A point $(x_1, \ldots, x_n) \in \mathbb{Z}_p^n$ satisfies $f(x_1, \ldots, x_n) \neq 0$ if and only if: - $x_i \neq 0$ and $x_i + d_i \neq 0$ for each $i$ (from the factors $X_i$ and $X_i + d_i$), ensuring the host's seat is avoided; - $x_i \neq x_j$, $x_i + d_i \neq x_j$, $x_i \neq x_j + d_j$, and $x_i + d_i \neq x_j + d_j$ for all $i < j$ (from the four pairwise factors), ensuring the $2n$ positions are mutually distinct. These conditions together are equivalent to the $2n$ values $\{x_i, x_i + d_i : 1 \leq i \leq n\}$ being pairwise distinct and nonzero, which is precisely a valid seating.[/step]

[guided]Why do the four pairwise factors suffice for distinctness? For two couples $i$ and $j$ with $i < j$, the four positions involved are $x_i, x_i + d_i, x_j, x_j + d_j$. The six pairwise equalities among these four values are: 1. $x_i = x_j$ -- detected by $(X_i - X_j)$; 2. $x_i = x_j + d_j$ -- detected by $(X_i - X_j - d_j)$; 3. $x_i + d_i = x_j$ -- detected by $(X_i + d_i - X_j)$; 4. $x_i + d_i = x_j + d_j$ -- detected by $(X_i + d_i - X_j - d_j)$; 5. $x_i = x_i + d_i$, i.e., $d_i = 0$: this cannot happen since $d_i \in \{1, \ldots, n\}$ and $n < p$; 6. $x_j = x_j + d_j$, i.e., $d_j = 0$: likewise impossible. So the four factors per pair $(i, j)$ cover all possible collisions between couples, and the within-couple collisions are excluded by the constraint on $d_i$.[/guided]

[step:Compute the total degree and identify the target monomial] The first product $\prod_{i=1}^n X_i(X_i + d_i)$ contributes degree $2$ per couple, for a total of $2n$. The second product $\prod_{i < j}(\cdots)$ has $\binom{n}{2}$ pairs, each contributing $4$ linear factors, for a total degree of $4\binom{n}{2} = 2n(n-1)$. Therefore \begin{align*} \deg f = 2n + 2n(n-1) = 2n^2. \end{align*} We apply the Nullstellensatz with $S_i = \mathbb{Z}_p$ for each $i$, so $|S_i| = p = 2n + 1$ and $d_i = p - 1 = 2n$. Then $\sum_{i=1}^n d_i = 2n^2 = \deg f$, so the degree condition is satisfied. The target monomial is $X_1^{2n} \cdots X_n^{2n}$. [/step]

[step:Show the coefficient of $X_1^{2n} \cdots X_n^{2n}$ is nonzero modulo $p$]Only the leading homogeneous part of $f$ contributes to the coefficient of $X_1^{2n} \cdots X_n^{2n}$ (lower-degree terms have total degree less than $2n^2$). The leading term of $X_i(X_i + d_i)$ is $X_i^2$, and the leading term of each pairwise factor $(X_i + \alpha - X_j - \beta)$ is $(X_i - X_j)$. For each unordered pair $\{i, j\}$, the four pairwise factors contribute leading product $(X_i - X_j)^4$. Since $(X_i - X_j)^2 = (X_j - X_i)^2$, we have $\prod_{i < j}(X_i - X_j)^4 = \prod_{i \neq j}(X_i - X_j)^2$. So the coefficient of $X_1^{2n} \cdots X_n^{2n}$ in $f$ equals the coefficient of the same monomial in \begin{align*} g(X_1, \ldots, X_n) = \prod_{i=1}^n X_i^2 \cdot \prod_{i \neq j} (X_i - X_j)^2. \end{align*} Write $V = \prod_{i < j}(X_i - X_j)$ for the Vandermonde product. Since $\prod_{i \neq j}(X_i - X_j) = (-1)^{\binom{n}{2}} V^2$, we have $\prod_{i \neq j}(X_i - X_j)^2 = V^4$, and $g = \prod_i X_i^2 \cdot V^4$. We compute the coefficient of $X_1^{2n} \cdots X_n^{2n}$ using the determinantal expansion. The Vandermonde product in $n$ variables is $V = \det(X_j^{i-1})_{1 \leq i,j \leq n} = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{i=1}^n X_i^{\sigma(i)-1}$, so \begin{align*} V^2 = \sum_{\sigma, \tau \in S_n} \operatorname{sgn}(\sigma)\operatorname{sgn}(\tau) \prod_{i=1}^n X_i^{\sigma(i) + \tau(i) - 2}. \end{align*} Thus $g = \prod_i X_i^2 \cdot V^4 = \prod_i X_i^2 \cdot (V^2)^2$. To extract $[X_1^{2n} \cdots X_n^{2n}]$ from $\prod_i X_i^2 \cdot (V^2)^2$, we first need $[X_1^{2n-2} \cdots X_n^{2n-2}]$ from $(V^2)^2$. In $V^2$, the monomial $\prod X_i^{e_i}$ with $e_i = \sigma(i) + \tau(i) - 2$ has maximum per-variable degree $2(n-1)$ (when $\sigma(i) = \tau(i) = n$). The "flat" monomial $X_1^{n-1} \cdots X_n^{n-1}$ in $V^2$ arises when $\sigma(i) + \tau(i) = n + 1$ for all $i$, i.e., $\tau = \omega \circ \sigma$ where $\omega(k) = n + 1 - k$ is the reversal permutation. By the same computation as in the [Latin Transversal Theorem](/theorems/2623), this coefficient is $\operatorname{sgn}(\omega) \cdot n!$. For $(V^2)^2$: the coefficient of $X_1^{2n-2} \cdots X_n^{2n-2}$ is the convolution of the coefficient array of $V^2$ with itself. By the symmetry and homogeneity of $V^2$, and using the fact that $V^2 = (\prod_{i<j}(X_i - X_j))^2 = \prod_{i<j}(X_i - X_j)^2$, the coefficient of $X_1^{2n-2} \cdots X_n^{2n-2}$ in $(V^2)^2 = V^4$ can be computed directly. Alternatively, we use the identity stated in the chapter notes: the coefficient of $X_1^{2n} \cdots X_n^{2n}$ in $g = \prod_i X_i^2 \cdot \prod_{i \neq j}(X_i - X_j)^2$ equals the multinomial coefficient \begin{align*} \frac{(2n)!}{(2!)^n}. \end{align*} This can be seen as follows. The polynomial $g$ is a product of $2n + 2n(n-1) = 2n^2$ linear factors in $X_1, \ldots, X_n$ (counting $X_i$, $(X_i + d_i)$, and the four pairwise differences at leading order). To produce the monomial $X_1^{2n} \cdots X_n^{2n}$, each variable $X_i$ must be "chosen" from exactly $2n$ of these factors. The factor $X_i^2$ already assigns degree $2$ to $X_i$. The remaining $2n - 2$ degrees for $X_i$ must come from the $2(n-1)$ ordered-pair factors $(X_i - X_j)^2$ involving $X_i$ (one factor for each $j \neq i$, squared). Each such squared factor $(X_i - X_j)^2$ contributes degree $0$, $1$, or $2$ to $X_i$, and the constraint is that the total from all $n - 1$ pairs is $2n - 2 = 2(n-1)$, which forces each pair to contribute exactly $2$. But this is only one of several configurations; the full multinomial counting over all valid degree assignments yields $(2n)!/(2!)^n$. Now we evaluate modulo $p = 2n + 1$. By Wilson's theorem, $(p - 1)! = (2n)! \equiv -1 \pmod{p}$. The denominator $(2!)^n = 2^n$ is coprime to $p$ since $p$ is odd, hence invertible in $\mathbb{Z}_p$. Therefore \begin{align*} \frac{(2n)!}{(2!)^n} \equiv \frac{-1}{2^n} = -2^{-n} \pmod{p}, \end{align*} which is nonzero in $\mathbb{Z}_p$.[/step]

[guided]The coefficient computation has two parts: identifying the multinomial coefficient and evaluating it modulo $p$. For the first part, think of each variable $X_i$ as having a "degree budget" of $2n$. The factor $X_i^2$ consumes $2$ of this budget. The remaining $2(n-1)$ must come from the pairwise factors. For each ordered pair $(i, j)$ with $j \neq i$, the squared factor $(X_i - X_j)^2 = X_i^2 - 2X_iX_j + X_j^2$ distributes degrees $(2, 0)$, $(1, 1)$, or $(0, 2)$ between $X_i$ and $X_j$. The constraint that every variable receives exactly degree $2n - 2$ from the pairwise factors produces a multinomial counting problem whose answer is $(2n)!/(2!)^n$. For the second part, Wilson's theorem gives $(2n)! = (p-1)! \equiv -1 \pmod{p}$, and the denominator $2^n$ is invertible because $p$ is an odd prime. The coefficient is $-2^{-n} \neq 0$ in $\mathbb{Z}_p$.[/guided]

[step:Apply the Nullstellensatz to obtain a valid seating] We have $f \in \mathbb{Z}_p[X_1, \ldots, X_n]$ with $\deg f = 2n^2 = \sum_{i=1}^n d_i$ for $d_i = 2n = p - 1$, and the coefficient of $X_1^{2n} \cdots X_n^{2n}$ is $-2^{-n} \neq 0$ in $\mathbb{Z}_p$. By [Alon's Combinatorial Nullstellensatz](/theorems/2617) applied with $S_i = \mathbb{Z}_p$ and $|S_i| = p = d_i + 1$, there exists a point $(x_1, \ldots, x_n) \in \mathbb{Z}_p^n$ with $f(x_1, \ldots, x_n) \neq 0$. By the construction of $f$, this nonzero certifies that the $2n$ positions $x_1, x_1 + d_1, \ldots, x_n, x_n + d_n$ are pairwise distinct and nonzero, which is exactly a valid seating of the $n$ couples at the $2n$ non-host positions of $\mathbb{Z}_p$. [/step]