Math @ Duke

Publications [#243907] of Ezra Miller
Papers Published
 with Guo, A; Miller, E, Lattice point methods for combinatorial games,
Advances in Applied Mathematics, vol. 46 no. 14
(2011),
pp. 363378, ISSN 01968858 ([math.CO/0908.3473] [math.CO/1105.5420].) [repository], [doi]
(last updated on 2018/07/22)
Abstract: We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These encompass all heap games in a natural setting where the SpragueGrundy theorem for normal play manifests itself geometrically. We provide an algorithm to compute normal play strategies. The setting of lattice games naturally allows for misère play, where 0 is declared a losing position. Lattice games also allow situations where larger finite sets of positions are declared losing. Generating functions for sets of winning positions provide data structures for strategies of lattice games. We conjecture that every lattice game has a rational strategy: a rational generating function for its winning positions. Additionally, we conjecture that every lattice game has an affine stratification: a partition of its set of winning positions into a finite disjoint union of finitely generated modules for affine semigroups. This conjecture is true for normalplay squarefree games and every lattice game with finite misère quotient. © 2010 Elsevier Inc. All rights reserved.


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

