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Charts

The primary data that describe a chart U are:

S

a list of affine variables (generators of k[S]), (in the HTML output this field appears with the name VAR),

DEP

a list of polynomials from k[S] (the algebraic relations of S), k[U]= k[S]/ <DEP>,

DEP2

has role (and appears in the debugging and HTML outputs) only when the useDEP2 configuration variable is true. It is the list of invertible functions over the chart.

IND

a list of polynomials (representatives of elements from k[U]), that give rise to a system of regular parameters at every point of U,

PDER

a matrix of polynomials from k[S] (representatives of elements from k[U]), the partial derivatives of the generators of k[S] w.r.t. the IND members,

MAP

a list of polynomials from k[S], representatives of the images of the initial generators: x₁, ..., x_n in the current chart (blowing up map),

LMAP

the list of images (polynomial representatives from k[S]) of the k-algebra generators of the parent chart in the resolution tree (local blowing up map),

IMAP

a list of rational functions from k(S'), where S' is the S member of the initial chart; these are the images of the generators in S along the inverse of MAP (blowing down map).

TAGS

in case of resolutions (and not ideal principalizations) after each completed resolution a new list of 0s and 1s is added to TAGS where the 1s mark those elements of IND that define the nonsingular strict transform over the chart. TAGS can contain more than one such list when subsequent components are resolved using dlocext. This data is used by dlocsep.

SCS

a list of polynomials, generators of an ideal in k[U], which is supposed to be the sections of an ideal sheaf over the variety for which U is a covering chart. This data is not used in the resolution. When a desingularization is completed, the SCS entries of the charts in the covering of the final blowing up variety can be set up by the user or another procedure and then the global section computation routines use these data.

The following data is necessary to administrate the evolution of the chart during the resolution process and to provide auxiliary information for optimization strategies:

Sn

an integer, the next available index for the new variable x_Sn,

DEP2V

has role (and appears in the debugging and HTML outputs) only when the useDEP2 configuration variable is true. It holds the single variable which denotes the inverse of the product of the functions in DEP2 (the variable is also a member of S).

FOCUS

a list of polynomials from k[S], defining the subvariety of the chart not being covered by charts with smaller identifiers,

SUBST

a list of equations, the substitutions made so far in the chart with which the algorithm eliminated variables,

IDs

is a list of data building up the identifier of the chart, and providing additional information on the relation to the parent and to the children of the chart in the resolution tree. Its data members are:

1. The index of the chart in chartHistory, i.e. the major identification number of the chart.
2. The identifier of the parent of the chart converted to a maple name (this is used in the HTML exporter to generate links to the parent).
3. The minor identification number of the chart (currently unused).
4. The transformation identifier: it encodes the transformation which was performed on the chart, or gives the state of the chart: 0 - the chart contains a single resolved basic object, 1 - the transformation on the chart was blowing up, 2 - the operation on the chart was exchange, 3 - the transformation of the chart was cover.
5. The maximum value of the stratifying function over the chart, defined by the current state of the resolution problem (encoded in the basic objects). The function value is represented by nested lists of numeric data, and it is generated for a chart record by the function `desing/vstfn`.
6. A list of triples that correspond to the children of the chart in the chart tree. The first member of a triple is the major identifier (i.e. index in chartHistory) of the child, the second is
   • in case of blowing up, the index of the equation in IND, which becomes the defining equation of the exceptional divisor over the child,
   • in case of exchange the index of the IND-element which gets exchanged,
   • in case of cover, the equation of the function which becomes invertible over the child.
   The third member of the triple is a list which
   • contains pairs: (variable, index), encoding which new variable over the child corresponds to the proper transform of which defining equation (identified by the index to IND) of the blowing up center,
   • is empty in case of exchange,
   • contains pairs: (variable, equation), encoding which new variable over the child is the inverse of which equation.