The input of the algorithm is a (list of) polynomial(s) in the
variables x_{1}, ..., x_{n}, defining an
affine scheme Z, embedded into a nonsingular variety (or
chart) X in A^{n}. Since the program
needs to generate new variables, it uses the pattern
x_{i} 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 x_{i}
are reserved for the program.
The program computes an embedded desingularization f: Y >
X of Z, and produces a list of ndimensional
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
.. globalChartCounter1, 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 chartrecords (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: x_{1}, ...,
x_{n} 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 leafcharts 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 1tags
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 equiorder 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 lexmaximal 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 lexmaximal tuple).
The structure of the HTMLtree:
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 inputoutput, 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*x3^4x3^4*x9^2*x7^2

x10*x7

x7

x9

x10

DEP  empty 

MAP 
x1 
x10*x7*x3

x2 
x7*x3

x3 
x3

x4 
x9*x7*x3


IMAP 
x3 
x3

x7 
x2/x3

x9 
x4/x2

x10 
x1/x2



STACK 
WBO (dim: 4) 
J 
x10^3*x7+x7^2*x10x9^2
 2 
c 
2 
N  empty 
(E,a) 
E  a 
x3  3 
x7  1 
C 
empty 
TBO (dim: 4) 
J 
x10^3*x7+x7^2*x10x9^2
 2 
N  empty 
E 
x3 
x7 
EM 
x3 
C 
empty 
SBO (dim: 4) 
J 
x10^3*x7+x7^2*x10x9^2
 2 
N  empty 
H  empty 
E 
x7 
C 
empty 
WBO (dim: 3) 
J 
x10^3+x10*x7
 2 
c 
1 
N 
x9 
(E,a) 
E  a 
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 
H  empty 
E  empty 
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 
E  empty 
EM  empty 
C 
empty 
SBO (dim: 2) 
J 
x7
 1 
N 
x9 
x10 
H  empty 
E  empty 
C 
empty 
Blown up to: 
32.0 
33.0 
34.0 
x9 
x10 
x7 
 ID: 19.0 
Parent: 3.0

Tree 

