The returned polynomial is
, where the coefficients
are the solution of the (transposed)
Vandermonde system of linear equations
Let be the ground field,
the
variables and
a monomial ordering, then Loc
denotes the
localization of
with respect to the multiplicatively closed set
lp:
dp:
Dp:
wp(1,2,3):
Wp(1,2,3):
ls:
ds:
Ds:
ws(1,2,3):
Ws(1,2,3):
(dp(3), wp(1,2,3)):
(Dp(3), ds(3)):
(dp(3), a(1,2,3),dp(3)):
(a(1,2,3,4,5),Dp(3), ds(3)):
and
a monomial
ordering on
and
a monomial ordering on
. The product
ordering (or block ordering)
on
is the following:
or (
and
).
is called a normal form of
with
respect to
(note that such a function is not unique).
For a standard basis
of
the following holds:
if and only if
.
Let
be a homogeneous module, then the Hilbert function
of
(see below)
and the Hilbert function
of the leading module
coincide, i.e.,
.
Let M
be a graded module over
with
respect to weights
.
The Hilbert function of
,
, is defined (on the integers) by
Moreover, let
ini
. The pseudo remainder
of
with respect to
is
defined by the equality
with
and
minimal.
A set
is called triangular if
. Moreover, let
,
then
is called a triangular system, if
is a triangular set
such that
does not vanish on
.
is called irreducible if for every
there are no
,
,
such that
The main result on triangular sets is the following:
let
then there are irreducible triangular sets
such that
where
. Such a set
is called an irreducible characteristic series of
the ideal
.
Let
be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by
.
Let
be the local ring at the origin and
the Jacobian ideal of
.
A Milnor representative of defines a differentiable fibre bundle over the punctured disc with fibres of homotopy type of
-spheres.
The
-th cohomology bundle is a flat vector bundle of dimension
and carries a natural flat connection with covariant derivative
.
The monodromy operator is the action of a positively oriented generator of the fundamental group of the puctured disc on the Milnor fibre.
Sections in the cohomology bundle of moderate growth at
form a regular
-module
, the Gauss-Manin connection.
By integrating along flat multivalued families of cycles, one can consider fibrewise global holomorphic differential forms as elements of .
This factors through an inclusion of the Brieskorn lattice
in
.
The -module structure defines the V-filtration
on
by
.
The Brieskorn lattice defines the Hodge filtration
on
by
which comes from the mixed Hodge structure on the Milnor fibre.
Note that
.
The induced V-filtration on the Brieskorn lattice determines the singularity spectrum by
.
The spectrum consists of
rational numbers
such that
are the eigenvalues of the monodromy.
These spectral numbers lie in the open interval
, symmetric about the midpoint
.
The spectrum is constant under -constant deformations and has the following semicontinuity property:
The number of spectral numbers in an interval
of all singularities of a small deformation of
is greater or equal to that of f in this interval.
For semiquasihomogeneous singularities, this also holds for intervals of the form
.
Two given isolated singularities and
determine two spectra and from these spectra we get an integer.
This integer is the maximal positive integer
such that the semicontinuity holds for the spectrum of
and
times the spectrum of
.
These numbers give bounds for the maximal number of isolated singularities of a specific type on a hypersurface
of degree
:
such a hypersurface has a smooth hyperplane section, and the complement is a small deformation of a cone over this hyperplane section.
The cone itself being a
-constant deformation of
, the singularities are bounded by the spectrum of
.
Using the library gaussman.lib one can compute the monodromy, the V-filtration on , and the spectrum.
If the principal part of
is
-nondegenerate, one can compute the spectrum using the library spectrum.lib.
In this case, the V-filtration on
coincides with the Newton-filtration on
which allows to compute the spectrum more efficiently.
on a surface
, the geometric genus
The singularities of type
form a
-constant one parameter family given by
.
.
on a septic in
. But
.
Let
denote an
matrix with integral coefficients. For
, we define
to be the uniquely determined
vectors with nonnegative coefficients and disjoint support (i.e.,
or
for each component
) such that
. For
component-wise, let
denote the monomial
.
The ideal
The first problem in computing toric ideals is to find a finite
generating set: Let
be a lattice basis of
(i.e, a basis of the
-module). Then
A lattice basis
is again computed via the
LLL-algorithm. The saturation step is performed in the following way:
First note that
induces a positive grading w.r.t. which the ideal
Let be a homogeneous ideal w.r.t. the weighted reverse
lexicographical ordering with weight vector
and variable order
. Let
denote a Groebner basis of
w.r.t. to
this ordering. Then a Groebner basis of
is obtained by
dividing each element of
by the highest possible power of
.
From this fact, we can successively compute
This procedure involves Groebner basis computations. Actually, this
number can be reduced to at most
to
Groebner basis
computations. It needs no auxiliary variables, but a supplementary
precondition; namely, the existence of a vector without zero components
in the kernel of
.
The main idea comes from the following observation:
Let be an integer matrix,
a lattice basis of the
integer kernel of
. Assume that all components of
are
positive. Then
The algorithm starts by finding a lattice basis
of the
kernel of
such that
has no zero component. Let
be the set of indices
with
. Multiplying the components
of
and the columns
of
by
yields
a matrix
and a lattice basis
of the kernel of
that fulfill the assumption of the observation above. We are then able
to compute a generating set of
by applying the following
``variable flip'' successively to
:
Let be an elimination ordering for
. Let
be the matrix
obtained by multiplying the
-th column of
with
. Let
The IP problem is very hard; namely, it is NP-complete.
For the following discussion let (component-wise). We
consider
as a weight vector; because of its non-negativity,
can
be refined into a monomial ordering
. It turns out that we can
solve such an IP instance with the help of toric ideals:
First we assume that an initial solution (i.e.,
) is already known. We obtain the optimal solution
(i.e., with
minimal) by the following procedure:
Faugère,