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The resolution process is guided by the Villamayor stratification function, which is defined recursively over a chain of basic objects with decreasing dimension, taking the exceptional divisors of previous blowing ups into account in a special way. All this results quite a complex function and leads to similarly complex resolutions in some measures. Though there are speedups already available, the fact that there is no known complexity bound for the algorithm gives an intuition about the involvedness of the computation.

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 x12-x2x32 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:
x1 x2 x3

The image with exceptional divisor x2 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.

All the other examples are computed with the speedup technique turned on.

Further available examples:

x16+x26- x1x2x3

x16-x24+ x1 x2 x33

x14 - x12 + x22x32

(x12+x22+ x32-1)(x12+ (x2-1)2-1)

x3x12+ x3x22+ x33-x1x2

x12- x23x33+ x13x3- x13x2

4D examples:

[x2 x3 x42-x13, dualgraph], [x12 x2-x32 x4, dualgraph] [x12 x3 x4+x22 x3 x4+x12 x22, dualgraph] [x1 x22 x33-x46, dualgraph] [x1x23+x13x3-x32x42, dualgraph] [(x12+x22+x32-1)2 x42-x44, dualgraph]

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