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LATTICE POINTS IN VECTOR-DILATED ALGEBRAIC POLYTOPES
Keywords:
Ehrhart polynomial, vector dilation, irrational polytope, lattice pointsDOI:
https://doi.org/10.17654/0972555522010Abstract
We introduce the Ehrhart theory of algebraic cross-polytopes that undergo vector dilations, generalizing previous work of Borda on scalar dilations of algebraic cross-polytopes and Beck on vector dilations of rational simplices. In particular, for a given class of algebraic cross-polytopes and a dilation vector t dilating each facet, we show that the number of lattice points can be approximated by an explicitly given polynomial of t depending only on the polytope. As a result, we obtain a form of the Ehrhart-Macdonald reciprocity law for the leading term.
Received: June 20, 2021
Revised: August 23, 2021
Accepted: November 12, 2021
References
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B. Borda, Lattice points in algebraic cross-polytopes and simplices, Discrete Comput. Geom. 60(1) (2018), 145-169.
E. Ehrhart, Sur un probléme de géométrie diophantienne linéaire II, J. Reine Angew. Math. 227 (1967), 25-49 (in French).
M. Nechayeva and B. Randol, Asymptotics of weighted lattice point counts inside dilating polygons, Additive Number Theory, Springer, New York, 2010, pp. 287-301.
B. Randol, On the number of integral lattice-points in dilations of algebraic polyhedra, Internat. Math. Res. Notices 1997(6) (1997), 259-270.
M. M. Skriganov, On integer points in polygons, Ann. Inst. Fourier (Grenoble) 43(2) (1993), 313-323.
M. M. Skriganov, Ergodic theory on Diophantine approximations and anomalies in the lattice point problem, Invent. Math. 132(1) (1998), 1-72.
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