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  "body": "\n## Statement\n\nLet $\\mathbb{F}$ be a finite field of characteristic $p$.\nLet $f_1, \\dots, f_r \\in \\mathbb{F}[X_1, \\dots, X_n]$ be polynomials with total degrees $d_1, \\dots, d_r$, such that $n \u003e d_1 + \\dots + d_r$.\nThen the number of common zeros $(a_1, \\dots, a_n) \\in \\mathbb{F}^n$ of $f_1, \\dots, f_r$ is divisible by $p$.\n\n## Remarks\n\nA corollary is Chevalley's theorem: if the polynomials have no constant terms, then they have a common zero other than $(0, \\dots, 0)$.\nThe reason is that $(0, \\dots, 0)$ is a zero, and the number of zeros is a multiple of $p \\ge 2$.\nFor example, a quadratic form in $3$ or more variables over $\\mathbb{Z}/p\\mathbb{Z}$ has a nontrivial zero.\n\n## History\n\nEwald Warning and Claude Chevalley proved the results in 1935.\n\n## Explanations\n\n### L1 human\n\nWhen a system of polynomial equations over a finite field has many variables compared with the degrees, the solutions come in multiples of $p$.\nSo if there is one obvious solution, such as all zeros, there must be another.\n",
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