The affine-stratification conjecture for lattice games
A lattice game is a game played on a set of lattice points in a polyhedron, with a finite compatible rule set determining its winning and losing positions. Let denote the set of P-positions. An affine stratification is a partition
where each is a finitely generated module for an affine semigroup ; explicitly, for a finite set .
Affine-stratification conjecture. Every lattice game possesses an affine stratification.
Affine stratifications capture regular decompositions observed in examples, including normal-play squarefree games, but the conjecture remains unproved for arbitrary lattice games.
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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The affine stratification conjecture for lattice games
Let be the set of winning positions of a lattice game. An affine stratification of is an expression of as a finite union of translates of affine semigroups. The affine stratification conjecture. Every lattice game has an affine stratification. Such a description would give a finer structural form of the winning positions than rational generating functions; the source gives no evidence of a resolution.
source: Ezra Miller, “Theory and applications of lattice point methods for binomial ideals”, arXiv:1009.2823 (2010).
References
Primary source
Alan Guo and Ezra Miller, “Lattice point methods for combinatorial games”, arXiv:0908.3473 (2009).
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