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  "body": "\n## Statement\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $f \\in F[x_1, \\dots, x_n]$ be a [[polynomial in $n$ variables|07GNX8GCYP]].\nLet $S_1, \\dots, S_n$ be nonempty finite subsets of $F$ and let $g_i(x_i) = \\prod_{s \\in S_i} (x_i - s)$.\n\n1. If $f(s_1, \\dots, s_n) = 0$ for all $(s_1, \\dots, s_n) \\in S_1 \\times \\dots \\times S_n$, then there are polynomials $h_1, \\dots, h_n \\in F[x_1, \\dots, x_n]$ with $\\deg h_i \\le \\deg f - \\deg g_i$ and $f = \\sum_{i=1}^n h_i g_i$.\n2. Let $t_1, \\dots, t_n \\ge 0$ be integers with $\\abs{S_i} \u003e t_i$ for each $i$. Suppose that the [[degree|0P4JRMSWNH]] of $f$ is $t_1 + \\dots + t_n$ and that the coefficient of $x_1^{t_1} \\cdots x_n^{t_n}$ in $f$ is nonzero. Then there is $(s_1, \\dots, s_n) \\in S_1 \\times \\dots \\times S_n$ with $f(s_1, \\dots, s_n) \\ne 0$.\n\n## Remarks\n\nThe result and its name are due to Alon.\nFor part 1, reduce $f$ by replacing each power $x_i^{e}$ with $e \\ge \\abs{S_i}$ using $g_i(x_i) = 0$; the remainder has degree less than $\\abs{S_i}$ in each $x_i$ and still vanishes on $S_1 \\times \\dots \\times S_n$.\nA polynomial of that kind which vanishes on the whole grid is zero.\nThis follows by induction on $n$, since a nonzero polynomial in one variable of degree $d$ has at most $d$ roots.\nPart 2 follows from part 1.\nIf $f$ vanished on the grid, then $f = \\sum_i h_i g_i$ with the stated degree bounds.\nEvery monomial of top degree in $h_i g_i$ has $x_i$-degree at least $\\abs{S_i} \u003e t_i$, so $x_1^{t_1} \\cdots x_n^{t_n}$ could not appear in $f$.\n\nThe size condition is needed: for $n = 1$, $f = x_1(x_1 - 1)$, $t_1 = 2$ and $S_1 = \\set{0, 1}$, we have $\\abs{S_1} = t_1$ and $f$ vanishes on $S_1$.\n\nExample: the [[Cauchy–Davenport theorem|F7K7XFACSG]] follows from part 2.\nFor a prime $p$ and nonempty $A, B \\subseteq \\Z/p\\Z$ with $\\abs{A} + \\abs{B} - 1 \\le p$, suppose the [[sumset|XNBN31HQWB]] $A + B$ lies in a set $C$ of size $\\abs{A} + \\abs{B} - 2$.\nApply part 2 to $f(x, y) = \\prod_{c \\in C} (x + y - c)$ with $t_1 = \\abs{A} - 1$ and $t_2 = \\abs{B} - 1$: the coefficient of $x^{t_1} y^{t_2}$ is $\\binom{t_1 + t_2}{t_1}$, which is nonzero modulo $p$, yet $f$ vanishes on $A \\times B$, a contradiction.\n\n## Explanations\n\n### L1 human\n\nA polynomial that is not too big cannot vanish on a large grid of points.\nThe theorem makes \"not too big\" precise: look at one top-degree term and compare its exponents with the sizes of the sets along each axis.\nIf each set is larger than the matching exponent, the polynomial is nonzero somewhere on the grid.\n\n### L2 human\n\nThe method is to encode a combinatorial property as the vanishing of a polynomial on a grid, then exhibit a nonzero top coefficient.\nIt gives short proofs of results in additive number theory, such as Cauchy–Davenport, and of results on graph colouring and on hyperplane coverings of the cube.\nIt is a sharp version, for grids, of the fact that a nonzero polynomial cannot vanish everywhere on $F^n$ when $F$ is infinite.\n",
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