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Computing the Ehrhart quasi-polynomial of a rational simplex. (English) Zbl 1093.52009

To a rational polytope \(P \subset {\mathbb R}^d\) (i.e., the convex hull of finitely many points in \({\mathbb Q}^d\)), we associate the integer-point counting function \(L_P(t) := \# (tP \cap {\mathbb Z}^d)\), defined for positive integers \(t\). E. Ehrhart’s central theorem [C. R. Acad. Sci., Paris 254, 616–618 (1962; Zbl 0100.27601)] asserts that \(L_P\) is a quasi-polynomial in \(t\), i.e., \(L_P\) is of the form \[ L_P(t) = c_n(t) t^d + c_{ n-1 }(t) t^{ n-1 } + \cdots + c_1(t) \, t + c_0(t), \] where \(c_0, c_1, \dots, c_n\) are periodic functions of \(t\). If \(P\) is an integral polytope, i.e., the vertices of \(P\) are in \({\mathbb Z}^d\), then the period of \(c_0, c_1, \dots, c_n\) is one, i.e., \(L_P\) is a polynomial.
Regarding the computational complexity of \(L_P\), a fundamental theorem of A. I. Barvinok [Math. Oper. Res. 19, No. 4, 769–779 (1994; Zbl 0821.90085)] states that in fixed dimension, the rational generating function \(\sum_{ t \geq 0 } L_P(t) x^t\) can be computed in time polynomial in the input data of \(P\). (If the dimension is not fixed, it is already an NP-hard problem to check whether there is an integer point in \(P\), even if \(P\) is a rational simplex.) Barvinok’s theorem implies that for an integral \(d\)-simplex \(\Delta\), we can compute the first \(k\) coefficients of the Ehrhart polynomial \(L_\Delta\) in polynomial time if we fix \(k\) and let \(d\) vary.
In the paper under review, Barvinok extends this result to the rational case. More precisely, the main theorem is as follows: For a fixed integer \(k \geq 0\), there exists a polynomial-time algorithm that, given any integer \(d \geq k\), a rational simplex \(\Delta \subset {\mathbb R}^d\), and an integer \(j \geq 0\), computes the value of \(c_{ d-k } (j)\). The underlying algorithm is based on a structural result that relates \(c_{ d-k } (j)\) to volumes of sections of \(\Delta\) by affine lattice subspaces parallel to faces of \(\Delta\) of dimension \(\geq d-k\).

MSC:

52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
05A15 Exact enumeration problems, generating functions
68R05 Combinatorics in computer science

Software:

LattE
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References:

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