3,818 research outputs found
On the representation of polyhedra by polynomial inequalities
A beautiful result of Br\"ocker and Scheiderer on the stability index of
basic closed semi-algebraic sets implies, as a very special case, that every
-dimensional polyhedron admits a representation as the set of solutions of
at most polynomial inequalities. Even in this polyhedral case,
however, no constructive proof is known, even if the quadratic upper bound is
replaced by any bound depending only on the dimension.
Here we give, for simple polytopes, an explicit construction of polynomials
describing such a polytope. The number of used polynomials is exponential in
the dimension, but in the 2- and 3-dimensional case we get the expected number
.Comment: 19 pages, 4 figures; revised version with minor changes proposed by
the referee
Polynomial inequalities representing polyhedra
Our main result is that every n-dimensional polytope can be described by at
most (2n-1) polynomial inequalities and, moreover, these polynomials can
explicitly be constructed. For an n-dimensional pointed polyhedral cone we
prove the bound 2n-2 and for arbitrary polyhedra we get a constructible
representation by 2n polynomial inequalities.Comment: 9 page
Computing parametric rational generating functions with a primal Barvinok algorithm
Computations with Barvinok's short rational generating functions are
traditionally being performed in the dual space, to avoid the combinatorial
complexity of inclusion--exclusion formulas for the intersecting proper faces
of cones. We prove that, on the level of indicator functions of polyhedra,
there is no need for using inclusion--exclusion formulas to account for
boundary effects: All linear identities in the space of indicator functions can
be purely expressed using half-open variants of the full-dimensional polyhedra
in the identity. This gives rise to a practically efficient, parametric
Barvinok algorithm in the primal space.Comment: 16 pages, 1 figure; v2: Minor corrections, new example and summary of
algorithm; submitted to journa
Exploiting Polyhedral Symmetries in Social Choice
A large amount of literature in social choice theory deals with quantifying
the probability of certain election outcomes. One way of computing the
probability of a specific voting situation under the Impartial Anonymous
Culture assumption is via counting integral points in polyhedra. Here, Ehrhart
theory can help, but unfortunately the dimension and complexity of the involved
polyhedra grows rapidly with the number of candidates. However, if we exploit
available polyhedral symmetries, some computations become possible that
previously were infeasible. We show this in three well known examples:
Condorcet's paradox, Condorcet efficiency of plurality voting and in Plurality
voting vs Plurality Runoff.Comment: 14 pages; with minor improvements; to be published in Social Choice
and Welfar
Convergent Puiseux Series and Tropical Geometry of Higher Rank
We propose to study the tropical geometry specifically arising from
convergent Puiseux series in multiple indeterminates. One application is a new
view on stable intersections of tropical hypersurfaces. Another one is the
study of families of ordinary convex polytopes depending on more than one
parameter through tropical geometry. This includes cubes constructed by
Goldfarb and Sit (1979) as special cases.Comment: 32 pages, 3 figure
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