Convex and discrete geometry Polytopes and polyhedra Theorem

g-theorem

Builds on f-vector and h-vector Dehn–Sommerville equations Binomial coefficient

Statement

Let $d \ge 1$ and let $(h_0, h_1, \dots, h_d)$ be a sequence of integers. Let $g_0 = h_0$ and $g_i = h_i - h_{i-1}$ for $1 \le i \le \floor{d/2}$ . Then $(h_0, \dots, h_d)$ is the h-vector of the boundary complex of a simplicial convex polytope of dimension $d$ if and only if the following hold:

$h_i = h_{d-i}$ for $0 \le i \le d$ .
$g_0 = 1$ .
$0 \le g_{i+1} \le g_i^{\langle i \rangle}$ for $1 \le i \le \floor{d/2} - 1$ , and $g_1 \ge 0$ .

Here, 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 $n_i > n_{i-1} > \dots > n_j \ge j \ge 1$ (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 $0^{\langle i \rangle} = 0$ .

Remarks

Condition 1 is the Dehn–Sommerville equations. Conditions 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 $h_0 \le h_1 \le \dots \le h_{\floor{d/2}}$ . The conditions were conjectured by McMullen. Billera and Lee proved sufficiency by constructing polytopes, and Stanley proved necessity with the hard Lefschetz theorem for toric varieties.

Example: the boundary of the $4$ -dimensional cross-polytope has $h = (1, 4, 6, 4, 1)$ and $g = (1, 3, 2)$ . Here $3 = \binom{3}{1}$ , so $3^{\langle 1 \rangle} = \binom{4}{2} = 6 \ge 2$ , and all conditions hold.

The sequence $(1, 1, 3, 1, 1)$ is symmetric, but $g = (1, 0, 2)$ and $0^{\langle 1 \rangle} = 0 < 2$ , so it is not the h-vector of a simplicial $4$ -polytope.

Going deeper

For a student

The g-theorem is a model of how commutative algebra and algebraic geometry solve combinatorial problems. The h-vector of a simplicial polytope is the Hilbert function of an Artinian reduction of its Stanley–Reisner ring. The hard Lefschetz property gives an injective multiplication map, and this forces the g-vector to be an M-sequence. The same characterization for all simplicial spheres was a long-standing conjecture; Adiprasito announced a proof in 2018.

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Tag	`4CZMWQVC30`
URI	`m2://4CZMWQVC30`
Kind	statement, theorem
Writing	W0 stub
MSC	52B05, 05E45
Status	This copy of the site has no build data. Run `m2 status 4CZMWQVC30`.
About	f-vector and h-vector
Uses	f-vector and h-vector, Dehn–Sommerville equations, Binomial coefficient
Changed	2 November 2026