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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]], let $n \\ge 0$, let $x_0, \\dots, x_n \\in K$ be pairwise distinct, and let $y_0, \\dots, y_n \\in K$.\nAn interpolation polynomial of the data $(x_i, y_i)$ is a polynomial $p \\in K[x]$ such that:\n\n1. $p = 0$ or $\\deg p \\le n$;\n2. $p(x_i) = y_i$ for $i = 0, \\dots, n$.\n\n## Remarks\n\nThere is exactly one interpolation polynomial.\nUniqueness: the difference of two of them has degree at most $n$ and $n + 1$ roots, so it is $0$ by the [[bound on the number of roots|0KJVQV3E0F]].\nExistence: the [[Lagrange formula|237KR6FRCK]] gives one.\nEquivalently, the coefficients solve a linear system whose matrix has the nonzero [[Vandermonde determinant|0P84B554AS]].\nExample: for $(0, 1)$, $(1, 3)$, $(2, 7)$ the interpolation polynomial is $x^2 + x + 1$.\nNon-example: $x^2 + x + 1 + x(x - 1)(x - 2)$ takes the same values at $0, 1, 2$, but it has degree $3 \u003e 2$, so clause 1 fails.\n\n## Explanations\n\n### L1 human\n\nThe degree bound is what makes the answer unique.\nWithout it, you can add any multiple of $(x - x_0) \\cdots (x - x_n)$ and still pass through the points.\n",
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