178 research outputs found
A Polyhedral Homotopy Algorithm For Real Zeros
We design a homotopy continuation algorithm, that is based on numerically
tracking Viro's patchworking method, for finding real zeros of sparse
polynomial systems. The algorithm is targeted for polynomial systems with
coefficients satisfying certain concavity conditions. It operates entirely over
the real numbers and tracks the optimal number of solution paths. In more
technical terms; we design an algorithm that correctly counts and finds the
real zeros of polynomial systems that are located in the unbounded components
of the complement of the underlying A-discriminant amoeba.Comment: some cosmetic changes are done and a couple of typos are fixed to
improve readability, mathematical contents remain unchange
The Higher Cicho\'n Diagram
For a strongly inacessible cardinal , we investigate the
relationships between the following ideals:
- the ideal of meager sets in the -box product topology
- the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799)
- the ideal of nowhere stationary subsets of a (naturally defined) stationary
set .
In particular, we analyse the provable inequalities between the cardinal
characteristics for these ideals, and we give consistency results showing that
certain inequalities are unprovable.
While some results from the classical case () can be easily
generalized to our setting, some key results (such as a Fubini property for the
ideal of null sets) do not hold; this leads to the surprising inequality
cov(null)non(null). Also, concepts that did not exist in the classical
case (in particular, the notion of stationary sets) will turn out to be
relevant.
We construct several models to distinguish the various cardinal
characteristics; the main tools are iterations with -support
(and a strong "Knaster" version of -cc) and one iteration with
-support (and a version of -properness).Comment: 84 page
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