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To see what the algorithm does in general, without going into the
subtle details, let's look at the first step of the resolution of the
surface defined by
*x _{1}^{2}-x_{2}x_{3}^{2}*
in affine three-space (the full resolution can be found among the
examples). The algorithm, according to the stratification function,
first blows up the origin, after creating nine basic objects in three
layers (for dimension 3, 2 and 1).

The input of blowing up. | ||

Proper transforms in the charts with exceptional divisors: | ||

x_{1} |
x_{2} |
x_{3} |

The image with exceptional divisor *x _{2}*
shows the same situation as the input problem, seemingly leading to
infinite cycle. On the other hand, the next blowing up center will be
determined by not only the singular locus of the variety, but also
the `history' of exceptional divisors.

The resolution then continues in all of the new charts, leading to new blowing ups or restrictions to affine open subsets, to finally reach a resolved state. A property of the Villamayor stratification function is that the resolved charts patch together to form the smooth variety of the computed resolution.

In the latest version of the package there is an optional computation
technique which slightly modifies Villamayor's stratifying function in
order to achieve simpler resolutions. This feature can be turned on
and off via the `usenctest` configuration variable. To
demonstrate its simplifying effect we present the resolution of the
above surface with and without using this strategy.

Further available examples:

x_{1}^{6}+x_{2}^{6}-
x_{1}x_{2}x_{3}dualgraph |
x_{1}^{6}-x_{2}^{4}+
x_{1} x_{2} x_{3}^{3}dualgraph |
x_{1}^{4} - x_{1}^{2} +
x_{2}^{2}x_{3}^{2}dualgraph |

(x_{1}^{2}+x_{2}^{2}+
x_{3}^{2}-1)(x_{1}^{2}+
(x_{2}-1)^{2}-1)dualgraph |
x_{3}x_{1}^{2}+
x_{3}x_{2}^{2}+
x_{3}^{3}-x_{1}x_{2}dualgraph |
x_{1}^{2}-
x_{2}^{3}x_{3}^{3}+
x_{1}^{3}x_{3}-
x_{1}^{3}x_{2}dualgraph |

4D examples:

[x_{2} x_{3} x_{4}^{2}-x_{1}^{3}, dualgraph],
[x_{1}^{2} x_{2}-x_{3}^{2} x_{4}, dualgraph]
[x_{1}^{2} x_{3} x_{4}+x_{2}^{2} x_{3} x_{4}+x_{1}^{2} x_{2}^{2}, dualgraph]
[x_{1} x_{2}^{2} x_{3}^{3}-x_{4}^{6}, dualgraph]
[x_{1}x_{2}^{3}+x_{1}^{3}x_{3}-x_{3}^{2}x_{4}^{2}, dualgraph]
[(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1)^{2} x_{4}^{2}-x_{4}^{4}, dualgraph]