Lattice point methods for combinatorial games

@article{Guo2011LatticePM,
  title={Lattice point methods for combinatorial games},
  author={Alan J. X. Guo and Ezra Miller},
  journal={Adv. Appl. Math.},
  year={2011},
  volume={46},
  pages={363-378}
}
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References

SHOWING 1-10 OF 51 REFERENCES
Algorithms for lattice games
TLDR
This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games using the theory of short rational generating functions.
An Investigation of Partizan Misere Games
TLDR
Partizan combinatorial \mis play games are investigated, by taking Plambeck's indistinguishability and \mis monoid theory for impartial positions and extending it to partizan ones, as well as examining the difficulties in constructing a category of \ Mis play games in a similar manner to Joyal's category of normal play games.
Short rational generating functions for lattice point problems
Abstract. We prove that for any fixed d the generating function of the projectionof the set of integer points in a rational d-dimensional polytope can be computed inpolynomial time. As a corollary, we
Combinatorial Games: Tic-Tac-Toe Theory
Preface A summary of the book in a nutshell Part I. Weak Win and Strong Draw: 1. Win vs. weak win 2. The main result: exact solutions for infinite classes of games Part II. Basic Potential Technique
The G-values of various games
A disjunctive combination of a finite set of two-person games Γ 1 , Γ 2 , …, Γ k may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual
Affine stratifications from finite misère quotients
TLDR
The motivating consequence of the main result is a special case of a conjecture due to Guo and the author on the existence of affine stratifications for (the set of winning positions of) any lattice game.
On Numbers and Games
ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally
Taming the wild in impartial combinatorial games
We introduce a misere quotient semigroup construction in impartial combinatorial game theory, and argue that it is the long-sought natural generalization of the normal-play Sprague-Grundy theory to
Misère Games and Misère Quotients
These notes are based on a short course offered at the Weizmann Institute of Science in Rehovot, Israel, in November 2006. The notes include an introduction to impartial games, starting from the
The complexity of generating functions for integer points in polyhedra and beyond
Motivated by the formula for the sum of the geometric series, we consider various classes  of sets S �¼ Zd of integer points for which an a priori �glong�h Laurent series or polynomial m�¸S xm can
...
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