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Embedding Divisor and Semi-Prime Testability in f-vectors of polytopes.
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In this article, it was shown that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable under standard conjectures on the density of primes and on $P\neq NP.Abstract:
We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The regime where we prove this computational difference (conditioned on standard conjectures on the density of primes and on $P\neq NP$) is when the dimension $d$ tends to infinity and the number of facets is linear in $d$.read more
References
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Journal ArticleDOI
The number of faces of a simplicial convex polytope
TL;DR: In this paper, the necessity of McMullen's condition on the f-vector of a simplicial convex d-polytope was shown to be complete and sufficient for f = (f., fi,..., fd...J of integers).
Journal ArticleDOI
A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes
Louis J. Billera,Carl W Lee +1 more
TL;DR: A proof of the sufficiency of McMullen's conditions for the characterization of the set of all f-vectors of simplicial convex d-polytopes is given.
Journal ArticleDOI
On the order of magnitude of the difference between consecutive prime numbers
Journal ArticleDOI
The numbers of faces of simplicial polytopes
TL;DR: In this article, the authors considered the problem of determining the possible f-vectors of simplicial polytopes and proved a conjecture about the form of the sclution to this problem.
Harald cram er and the distribution of prime numbers
TL;DR: In this article, the authors start with the one mathematical proof that every mathematician and statis- tician knows, Euclid's proof of the innitude, and then the eighties when the hitherto seemingly solid bedrock of heuristic and conjecture was shattered by a short, brilliant paper of Maier; and now, the nineties, when we are picking up the pieces, trying to make sense of what we now know.