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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $(h_0, h_1, \\dots, h_d)$ be a sequence of integers.\nLet $g_0 = h_0$ and $g_i = h_i - h_{i-1}$ for $1 \\le i \\le \\floor{d/2}$.\nThen $(h_0, \\dots, h_d)$ is the [[h-vector|DCK64GFPD7]] of the boundary complex of a simplicial convex polytope of dimension $d$ if and only if the following hold:\n\n1. $h_i = h_{d-i}$ for $0 \\le i \\le d$.\n2. $g_0 = 1$.\n3. $0 \\le g_{i+1} \\le g_i^{\\langle i \\rangle}$ for $1 \\le i \\le \\floor{d/2} - 1$, and $g_1 \\ge 0$.\n\nHere, for integers $a \\ge 0$ and $i \\ge 1$, write $a = \\binom{n_i}{i} + \\binom{n_{i-1}}{i-1} + \\dots + \\binom{n_j}{j}$ with $n_i \u003e n_{i-1} \u003e \\dots \u003e n_j \\ge j \\ge 1$ (this expansion exists and is unique), and set $a^{\\langle i \\rangle} = \\binom{n_i + 1}{i + 1} + \\binom{n_{i-1} + 1}{i} + \\dots + \\binom{n_j + 1}{j + 1}$, with $0^{\\langle i \\rangle} = 0$.\n\n## Remarks\n\nCondition 1 is the [[Dehn–Sommerville equations|4WKHF5BCTA]].\nConditions 2 and 3 say that $(g_0, \\dots, g_{\\floor{d/2}})$ is an M-sequence, the Hilbert function of a graded quotient of a polynomial ring generated in degree $1$; in particular $h_0 \\le h_1 \\le \\dots \\le h_{\\floor{d/2}}$.\nThe conditions were conjectured by McMullen.\nBillera and Lee proved sufficiency by constructing polytopes, and Stanley proved necessity with the hard Lefschetz theorem for toric varieties.\n\nExample: the boundary of the $4$-dimensional cross-polytope has $h = (1, 4, 6, 4, 1)$ and $g = (1, 3, 2)$.\nHere $3 = \\binom{3}{1}$, so $3^{\\langle 1 \\rangle} = \\binom{4}{2} = 6 \\ge 2$, and all conditions hold.\n\nThe sequence $(1, 1, 3, 1, 1)$ is symmetric, but $g = (1, 0, 2)$ and $0^{\\langle 1 \\rangle} = 0 \u003c 2$, so it is not the h-vector of a simplicial $4$-polytope.\n\n## Explanations\n\n### L1 human\n\nThe theorem lists exactly which face counts a polytope with simplex faces can have.\nBesides the mirror symmetry of Dehn–Sommerville, the h-vector must rise up to its middle, and it cannot rise too fast.\n\"Too fast\" is measured by a precise rule from commutative algebra due to Macaulay.\n\n### L2 human\n\nThe g-theorem is a model of how commutative algebra and algebraic geometry solve combinatorial problems.\nThe h-vector of a simplicial polytope is the Hilbert function of an Artinian reduction of its Stanley–Reisner ring.\nThe hard Lefschetz property gives an injective multiplication map, and this forces the g-vector to be an M-sequence.\nThe same characterization for all simplicial spheres was a long-standing conjecture; Adiprasito announced a proof in 2018.\n",
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