# Input-Output Descriptions (Maple version)

The input of the algorithm is a (list of) polynomial(s) in the variables x1, ..., xn, defining an affine scheme Z, embedded into a nonsingular variety (or chart) X in An. Since the program needs to generate new variables, it uses the pattern xi for i>n. In order to avoid name collisions, the user has to obey the caveat that the input is always in the form above, and also the names xi are reserved for the program.

The program computes an embedded desingularization f: Y -> X of Z, and produces a list of n-dimensional affine charts, that provide open covering for the variety Y. The full resolution, i.e. the complete list of charts (including the ones that are produced in the intermediate stages of the resolution), is available in the global array chartHistory (at indices 0 .. globalChartCounter-1, where the latter name is also a global variable).

While the above output is provided for other algorithms to perform further computations, the HTML export of the resolution is intended for humans. Here is the summary of the data of chart-records (collected data of charts and basic objects) that are provided in the HTML output:

A chart U contains the following data:

VAR
a list of affine variables (generators of k[VAR]),
DEP
a list of polynomials from k[VAR] (the algebraic relations of VAR), k[U]= k[VAR]/< DEP >.
IND
a list of polynomials (representatives of elements from k[U], give rise to a system of regular parameters in every point of U),
PDER
a matrix of polynomials from k[VAR](being representatives of elements from k[U]), storing the partial derivatives of the generators of k[VAR] w.r.t. the IND members,
FOCUS
a list of polynomials from k[VAR], defining the subset of the chart which is not covered by charts with smaller index,
MAP
a list of polynomials from k[VAR], the images of the initial generators: x1, ..., xn in k[U].
IMAP
a list of rational functions from k(VAR'), where VAR' is the VAR member of the initial chart; these are the images of the generators of k[U] along the inverse of MAP.
TAGS
a list of lists of zeroes and ones tagging the IND entry. if an algebraic set is resolved, in the leaf-charts its strict transform can be defined by IND elements. Then, at the end of the resolution of the algebraic set a list that marks with 1-tags the defining IND elements is appended to TAGS. This has real usage in desingext and desingsep. This entry is nonempty only if there is already a resolution completed.
When the useDEP2 configuration variable is true, i.e. when the algorithm uses a single indeterminate to represent the inverse for (the product of) all the invertible functions over the chart, additionally
DEP2
a list of polynomials from k[VAR], the invertible functions over the chart.
DEP2V
an element from VAR, the indeterminate that represents the inverse of the product of the functions in DEP2.
The stack contains weighted basic objects (WBO), tagged basic objects (TBO) and simple basic objects (SBO). Each TBO is created to have its singular locus equal to the equi-order locus of the previous WBO. Then each SBO is created to have the singular locus of the previous TBO restricted by the intersection of the maximal number of hypersurfaces from the EM entry of the TBO, whose intersection with the singular locus is not empty (ties are broken by choosing lex-maximal hypersurface tuples). Each WBO (except the one with largest dimension) is created to be one dimension less than the previous SBO (by coefficient object computation).

A weighted basic object contains the following data:

N
a list of pairwise different IND elements (practically indices to IND) defining the ambient variety W of the basic object,
J
the generators for the ideal of the basic object, polynomials from k[VAR], with rational exponents,
c
a rational number; the weighted order,
E
the list of equations of exceptional divisors,
a
the weights of the exceptional divisors,
C
a list of functions from IND to the rational numbers; a history of J modificators (defining a summand to J).
A tagged basic object contains the same data as a WBO, except the entries c and a, plus the entry:
EM
a list of equations of exceptional divisors (the E- history).
A simple basic object contains the same data as a WBO, except the entries c and a, plus the entry:
H
a list of equations of exceptional divisors; the maximal length list of elements from the EM entry of the previous TBO, whose intersection with the singular locus of the TBO is not empty (taking the lex-maximal tuple).
The structure of the HTML-tree:

The charts form the tree of the resolution. The HTML output contains the vertices (i.e. charts) at their states before one of the three main operations (blowup, exchange, cover). In the tree the children of a vertex appear right below it. Please note that chronological information for a chart should not be inferred from the depth at which it appears in the tree, but by comparing its stratifying function value (displayed by default for each nodes) to other charts. Also please remember that the stratifying function is completely computed only at blowing ups (red vertices), while for other kind of vertices (green and blue) the printed stratifying function value is only partial.

For more information on the input-output, data structures, features and configuration of the package, please consult the manual.

ID: 19.0 Parent: 3.0 Tree
Blown up to:
32.0 33.0 34.0
x9 x10 x7
PDER
VAR\IND x10 x7 x3 x9
x3 0 0 1 0
x7 0 1 0 0
x9 0 0 0 1
x10 1 0 0 0
FOCUS
x10*x7^4*x3^4+x10^3*x7^3*x3^4-x3^4*x9^2*x7^2
x10*x7
x7
x9
x10
DEPempty
MAP
x1 x10*x7*x3 x7*x3 x3 x9*x7*x3
IMAP
x3 x3 x2/x3 x4/x2 x1/x2
STACK
WBO (dim: 4)
J
x10^3*x7+x7^2*x10-x9^2 2
c 2
Nempty
(E,a)
Ea
x3 3
x7 1
C empty
TBO (dim: 4)
J
x10^3*x7+x7^2*x10-x9^2 2
Nempty
E
x3
x7
EM
x3
C empty
SBO (dim: 4)
J
x10^3*x7+x7^2*x10-x9^2 2
Nempty
Hempty
E
x7
C empty
WBO (dim: 3)
J
x10^3+x10*x7 2
c 1
N
x9
(E,a)
Ea
x7 1/2
C empty
TBO (dim: 3)
J
x10^3+x10*x7 2
N
x9
E
x7
EM
x7
C empty
SBO (dim: 3)
J
x10^3+x10*x7 2
N
x9
Hempty
Eempty
C empty
WBO (dim: 2)
J
x7 1
c 1
N
x9
x10
(E,a)empty
C empty
TBO (dim: 2)
J
x7 1
N
x9
x10
Eempty
EMempty
C empty
SBO (dim: 2)
J
x7 1
N
x9
x10
Hempty
Eempty
C empty
Blown up to:
32.0 33.0 34.0
x9 x10 x7
ID: 19.0 Parent: 3.0 Tree

Remark: This description applies to the HTML output of version 1.3 of desing.