955 research outputs found
The number of ramified coverings of the sphere by the double torus, and a general form for higher genera
An explicit expression is obtained for the generating series for the number
of ramified coverings of the sphere by the double torus, with elementary branch
points and prescribed ramification type over infinity. Thus we are able to
prove a conjecture of Graber and Pandharipande, giving a linear recurrence
equation for the number of these coverings with no ramification over infinity.
The general form of the series is conjectured for the number of these coverings
by a surface of arbitrary genus that is at least two.Comment: 14pp.; revised version has two additional results in Section
On Sharing, Memoization, and Polynomial Time (Long Version)
We study how the adoption of an evaluation mechanism with sharing and
memoization impacts the class of functions which can be computed in polynomial
time. We first show how a natural cost model in which lookup for an already
computed value has no cost is indeed invariant. As a corollary, we then prove
that the most general notion of ramified recurrence is sound for polynomial
time, this way settling an open problem in implicit computational complexity
General Ramified Recurrence is Sound for Polynomial Time
Leivant's ramified recurrence is one of the earliest examples of an implicit
characterization of the polytime functions as a subalgebra of the primitive
recursive functions. Leivant's result, however, is originally stated and proved
only for word algebras, i.e. free algebras whose constructors take at most one
argument. This paper presents an extension of these results to ramified
functions on any free algebras, provided the underlying terms are represented
as graphs rather than trees, so that sharing of identical subterms can be
exploited
Counting problem on wind-tree models
We study periodic wind-tree models, billiards in the plane endowed with
-periodically located identical connected symmetric right-angled
obstacles. We show asymptotic formulas for the number of (isotopy classes of)
closed billiard trajectories (up to -translations) on the
wind-tree billiard. We also compute explicitly the associated Siegel-Veech
constant for generic wind-tree billiards depending on the number of corners on
the obstacle.Comment: 41 pages, 15 figures. arXiv admin note: substantial text overlap with
arXiv:1502.06405 by other author
Finitely ramified iterated extensions
Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of
degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold
iterate of f, is absolutely irreducible over F; we compute a recursion for its
discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in
a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the
iterated monodromy group of f. The iterated extension L/F is finitely ramified
if and only if f is post-critically finite (pcf). We show that, moreover, for
pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely
ramified over K, pointing to the possibility of studying Galois groups with
restricted ramification via tree representations associated to iterated
monodromy groups of pcf polynomials. We discuss the wildness of ramification in
some of these representations, describe prime decomposition in terms of certain
finite graphs, and also give some examples of monogene number fields.Comment: 19 page
A proof of a conjecture for the number of ramified coverings of the sphere by the torus
An explicit expression is obtained for the generating series for the number
of ramified coverings of the sphere by the torus, with elementary branch points
and prescribed ramification type over infinity. This proves a conjecture of
Goulden, Jackson and Vainshtein for the explicit number of such coverings.Comment: 10 page
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