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This function can be called when there is already a completed resolution stored in chartHistory.

a polynomial from k[VAR], defining the scheme to be desingularized, where VAR is the list of variables of the root chart in the resolution tree stored in chartHistory.
The value of the resolvedChartList global variable, containing the data of the chart covering the final blowing up variety.
The effect is that the strict transform of J is computed along the map of the resolution which can be found in chartHistory and new basic objects are set up over the leaf-charts of the tree. Then the objects are resolved, such that the existing resolution is "extended" with the resolution of the strict transform of J.
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