11,020 research outputs found
The vanishing ideal of a finite set of points with multiplicity structures
Given a finite set of arbitrarily distributed points in affine space with
arbitrary multiplicity structures, we present an algorithm to compute the
reduced Groebner basis of the vanishing ideal under the lexicographic ordering.
Our method discloses the essential geometric connection between the relative
position of the points with multiplicity structures and the quotient basis of
the vanishing ideal, so we will explicitly know the set of leading terms of
elements of I. We split the problem into several smaller ones which can be
solved by induction over variables and then use our new algorithm for
intersection of ideals to compute the result of the original problem. The new
algorithm for intersection of ideals is mainly based on the Extended Euclidean
Algorithm.Comment: 12 pages,12 figures,ASCM 201
Multivariable Hodge theoretical invariants of germs of plane curves
We describe methods for calculation of polytopes of quasiadjunction for plane
curve singularities which are invariants giving a Hodge theoretical refinement
of the zero sets of multivariable Alexander polynomials. In particular we
identify some hyperplanes on which all polynomials in multivariable Bernstein
ideal vanish
A Jacobian module for disentanglements and applications to Mond's conjecture
Given a germ of holomorphic map from to ,
we define a module whose dimension over is an upper bound
for the -codimension of , with equality if is weighted
homogeneous. We also define a relative version of the module, for
unfoldings of . The main result is that if are nice
dimensions, then the dimension of over is an upper bound of
the image Milnor number of , with equality if and only if the relative
module is Cohen-Macaulay for some stable unfolding . In particular,
if is Cohen-Macaulay, then we have Mond's conjecture for .
Furthermore, if is quasi-homogeneous, then Mond's conjecture for is
equivalent to the fact that is Cohen-Macaulay. Finally, we observe
that to prove Mond's conjecture, it suffices to prove it in a suitable family
of examples.Comment: 19 page
On Hodge spectrum and multiplier ideals
We describe a relation between two invariants that measure the complexity of
a hypersurface singularity. One is the Hodge spectrum which is related to the
monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The
other is the multiplier ideal, having to do with log resolutions.Comment: shorter final version to appear in Math. An
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