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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 P\mathcal{P} denote the set of P-positions. An affine stratification is a partition

P=⨄i=1rWi\mathcal{P}=\biguplus_{i=1}^r W_i

where each WiW_i is a finitely generated module for an affine semigroup Ai⊂ZdA_i\subset\mathbb{Z}^d; explicitly, Wi=Fi+AiW_i=F_i+A_i for a finite set Fi⊂ZdF_i\subset\mathbb{Z}^d.

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.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The affine stratification conjecture for lattice games

    Let WW be the set of winning positions of a lattice game. An affine stratification of WW is an expression of WW 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).

Sources & referencesView supporting material

Primary source

Alan Guo and Ezra Miller, “Lattice point methods for combinatorial games”, arXiv:0908.3473 (2009).

affine-semigroupscombinatorial-gameslattice-gameslattice-points

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