2,522 research outputs found
Accelerating Parametric Probabilistic Verification
We present a novel method for computing reachability probabilities of
parametric discrete-time Markov chains whose transition probabilities are
fractions of polynomials over a set of parameters. Our algorithm is based on
two key ingredients: a graph decomposition into strongly connected subgraphs
combined with a novel factorization strategy for polynomials. Experimental
evaluations show that these approaches can lead to a speed-up of up to several
orders of magnitude in comparison to existing approache
The computational complexity of the Chow form
We present a bounded probability algorithm for the computation of the Chow
forms of the equidimensional components of an algebraic variety. Its complexity
is polynomial in the length and in the geometric degree of the input equation
system defining the variety. In particular, it provides an alternative
algorithm for the equidimensional decomposition of a variety.
As an application we obtain an algorithm for the computation of a subclass of
sparse resultants, whose complexity is polynomial in the dimension and the
volume of the input set of exponents. As a further application, we derive an
algorithm for the computation of the (unique) solution of a generic
over-determined equation system.Comment: 60 pages, Latex2
Limits of PGL(3)-translates of plane curves, II
Every complex plane curve C determines a subscheme S of the of 3x3
matrices, whose projective normal cone (PNC) captures subtle invariants of C.
In "Limits of PGL(3)-translates of plane curves, I" we obtain a set-theoretic
description of the PNC and thereby we determine all possible limits of families
of plane curves whose general element is isomorphic to C. The main result of
this article is the determination of the PNC as a cycle; this is an essential
ingredient in our computation in "Linear orbits of arbitrary plane curves" of
the degree of the PGL(3)-orbit closure of an arbitrary plane curve, an
invariant of natural enumerative significance.Comment: 22 pages. Minor revision. Final versio
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
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