867,576 research outputs found
Matrix-F5 algorithms over finite-precision complete discrete valuation fields
Let be a sequence
of homogeneous polynomials with -adic coefficients. Such system may happen,
for example, in arithmetic geometry. Yet, since is not an
effective field, classical algorithm does not apply.We provide a definition for
an approximate Gr{\"o}bner basis with respect to a monomial order We
design a strategy to compute such a basis, when precision is enough and under
the assumption that the input sequence is regular and the ideals are weakly--ideals. The conjecture of Moreno-Socias
states that for the grevlex ordering, such sequences are generic.Two variants
of that strategy are available, depending on whether one lean more on precision
or time-complexity. For the analysis of these algorithms, we study the loss of
precision of the Gauss row-echelon algorithm, and apply it to an adapted
Matrix-F5 algorithm. Numerical examples are provided.Moreover, the fact that
under such hypotheses, Gr{\"o}bner bases can be computed stably has many
applications. Firstly, the mapping sending to the reduced
Gr{\"o}bner basis of the ideal they span is differentiable, and its
differential can be given explicitly. Secondly, these hypotheses allows to
perform lifting on the Grobner bases, from to
or Finally, asking for the same
hypotheses on the highest-degree homogeneous components of the entry
polynomials allows to extend our strategy to the affine case
Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves
Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive
epsilon, we want to compute an epsilon-isotopic polygonal approximation to the
restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on
subdivision algorithms because of their adaptive complexity and ease of
implementation. Plantinga and Vegter gave a numerical subdivision algorithm
that is exact when the curve S is bounded and non-singular. They used a
computational model that relied only on function evaluation and interval
arithmetic. We generalize their algorithm to any bounded (but possibly
non-simply connected) region that does not contain singularities of S. With
this generalization as a subroutine, we provide a method to detect isolated
algebraic singularities and their branching degree. This appears to be the
first complete purely numerical method to compute isotopic approximations of
algebraic curves with isolated singularities
Steganography: a class of secure and robust algorithms
This research work presents a new class of non-blind information hiding
algorithms that are stego-secure and robust. They are based on some finite
domains iterations having the Devaney's topological chaos property. Thanks to a
complete formalization of the approach we prove security against watermark-only
attacks of a large class of steganographic algorithms. Finally a complete study
of robustness is given in frequency DWT and DCT domains.Comment: Published in The Computer Journal special issue about steganograph
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