2,376 research outputs found
Randomness for Free
We consider two-player zero-sum games on graphs. These games can be
classified on the basis of the information of the players and on the mode of
interaction between them. On the basis of information the classification is as
follows: (a) partial-observation (both players have partial view of the game);
(b) one-sided complete-observation (one player has complete observation); and
(c) complete-observation (both players have complete view of the game). On the
basis of mode of interaction we have the following classification: (a)
concurrent (both players interact simultaneously); and (b) turn-based (both
players interact in turn). The two sources of randomness in these games are
randomness in transition function and randomness in strategies. In general,
randomized strategies are more powerful than deterministic strategies, and
randomness in transitions gives more general classes of games. In this work we
present a complete characterization for the classes of games where randomness
is not helpful in: (a) the transition function probabilistic transition can be
simulated by deterministic transition); and (b) strategies (pure strategies are
as powerful as randomized strategies). As consequence of our characterization
we obtain new undecidability results for these games
Bounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves
In this article we extend work of Shanks and Washington on cyclic extensions,
and elliptic curves associated to the simplest cubic fields. In particular, we
give families of examples of hyperelliptic curves defined over
, with of degree , where is a Sophie Germain prime,
such that the rank of the Mordell--Weil group of the jacobian of
is bounded by the genus of and the -rank of the class group of the
(cyclic) field defined by , and exhibit examples where this bound is
sharp.Comment: 22 pages, To appear in J. Th\'eor. Nombres Bordeau
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