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  "body": "\n## Definition\n\nLet $m \\ge 1$ and $n \\ge 0$, and let $x_1, \\dots, x_m \\in \\R$ be pairwise distinct.\nAn *incidence matrix* is an $m \\times (n + 1)$ matrix $E = (e_{ij})$, with $1 \\le i \\le m$ and $0 \\le j \\le n$.\nIts entries are $0$ or $1$, and exactly $n + 1$ of them are equal to $1$.\nGiven real numbers $y_{ij}$ for the pairs with $e_{ij} = 1$, the *Birkhoff interpolation problem* asks for a polynomial $p$ with real coefficients such that:\n\n1. $p = 0$ or $\\deg p \\le n$;\n2. $p^{(j)}(x_i) = y_{ij}$ for every pair $(i, j)$ with $e_{ij} = 1$, where $p^{(j)}$ is the $j$-th [[derivative|0V1EJCT8G6]].\n\nThe pair of $E$ and the nodes is *poised* if this problem has exactly one solution for every choice of the data $y_{ij}$.\n\n## Remarks\n\nThe conditions form a square linear system in the $n + 1$ coefficients of $p$, so poisedness means that its determinant is nonzero.\nIf in each row of $E$ the ones form an initial block $e_{i0} = \\cdots = e_{i,k_i - 1} = 1$, the problem is [[Hermite interpolation|9Z50M2WPTP]], which is always poised.\nIf moreover each row has a single one in column $0$, it is [[polynomial interpolation|SS1YMS2NPC]].\n\nExample: nodes $x_1 = 0$, $x_2 = 1$, $n = 1$, conditions $p(0) = a$ and $p'(1) = b$.\nFor $p = c_0 + c_1 x$ these give $c_0 = a$ and $c_1 = b$, so the problem is poised.\n\nNon-example: nodes $-1, 0, 1$, $n = 2$, conditions $p(-1) = a$, $p'(0) = b$, $p(1) = c$.\nFor $p = c_0 + c_1 x + c_2 x^2$ the system has the matrix with rows $(1, -1, 1)$, $(0, 1, 0)$, $(1, 1, 1)$, whose determinant is $0$.\nFor instance $p = x^2 - 1$ is a nonzero solution with zero data, so the problem is not poised.\n",
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