2,455 research outputs found
Beyond Sperner's lemma
The present paper is devoted to a recent beautiful and ingenious proof of
Brouwer's fixed point theorem due to mathematical economists H. Petri and M.
Voorneveld. The heart of this proof is an analogue of Sperner's lemma motivated
by Shapley-Scarf model of markets of agents with preferences over indivisible
goods.
The goal of the paper is to present a relatively abstract version of
Petri-Voorneveld proof which makes transparent both its similarities and its
differences with the classical proof based on Sperner's lemma and a well known
Knaster-Kuratowski-Mazurkiewich argument.Comment: 9 page
Scarf's theorems, simplices, and oriented matroids
In 1967 Herbert Scarf suggested a new proof of Brouwer fixed point theorem
based on a surprising analogue of Sperner's lemma. This analogue was motivated
by Scarf's work in game theory and mathematical economics. Moreover, Scarf
proved a much general version of Sperner's lemma dealing with colorings by
vectors. The present paper begins by revisiting Scarf's ideas from the point of
view of the basic theory of simplicial cochains in the spirit of author's
papers arXiv:1909.00940 and arXiv:2012.13104. After this we get to the main new
results of the paper, namely, to a generalization of Scarf results to colorings
with colors belonging to an oriented matroid. No knowledge of the theory of
oriented matroids is assumed. In the last section we return to the original
Scarf theorem and reprove it using even more classical methods of the
combinatorial topology of Euclidean spaces. Also, we generalize a theorem of
Kannai.Comment: 63 page
Balanced Simplices on Polytopes
The well known Sperner lemma states that in a simplicial subdivision of a simplex with a properly labeled boundary there is a completely labeled simplex. We present two combinatorial theorems on polytopes which generalize Sperner's lemma.Using balanced simplices, a generalized concept of completely labeled simplices, a uni ed existence result of balanced simplices in any simplicial subdivision of a polytope is given.This theorem implies the well-known lemmas of Sperner, Scarf, Shapley, and Garcia as well as some other results as special cases.A second theorem which imposes no restrictions on the integer labeling rule is established; this theorem implies several results of Freund.
Generic and Cogeneric Monomial Ideals
Monomial ideals which are generic with respect to either their generators or
irreducible components have minimal free resolutions derived from simplicial
complexes. For a generic monomial ideal, the associated primes satisfy a
saturated chain condition, and the Cohen-Macaulay property implies shellability
for both the Scarf complex and the Stanley-Reisner complex. Reverse
lexicographic initial ideals of generic lattice ideals are generic.
Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in
the cogeneric case the Cohen-Macaulay type is greater than or equal to the
number of irreducible components. Methods of proof include Alexander duality
and Stanley's theory of local h-vectors.Comment: 15 pages, LaTe
Monomials, Binomials, and Riemann-Roch
The Riemann-Roch theorem on a graph G is related to Alexander duality in
combinatorial commutive algebra. We study the lattice ideal given by chip
firing on G and the initial ideal whose standard monomials are the G-parking
functions. When G is a saturated graph, these ideals are generic and the Scarf
complex is a minimal free resolution. Otherwise, syzygies are obtained by
degeneration. We also develop a self-contained Riemann-Roch theory for artinian
monomial ideals.Comment: 18 pages, 2 figures, Minor revision
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