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{\bf \'Eric Fusy}
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{\bf Counting $d$-Polytopes with $d+3$ Vertices}
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We completely solve the problem of enumerating combinatorially
inequivalent $d$-dimensional polytopes with $d+3$ vertices. A first
solution of this problem, by Lloyd, was published in 1970. But the
obtained counting formula was not correct, as pointed out in the new
edition of Gr\"unbaum's book. We both correct the mistake of Lloyd and
propose a more detailed and self-contained solution, relying on
similar preliminaries but using then a different enumeration method
involving automata. In addition, we introduce and solve the problem of
counting oriented and achiral (i.e., stable under reflection)
$d$-polytopes with $d+3$ vertices. The complexity of computing tables
of coefficients of a given size is then analyzed. Finally, we derive
precise asymptotic formulas for the numbers of $d$-polytopes, oriented
$d$-polytopes and achiral $d$-polytopes with $d+3$ vertices. This
refines a first asymptotic estimate given by Perles.
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