Abstract: (with Gaffney and Trotman) This
paper will examine the $t^r$ conditions which were introduced by Thom and
Trotman, which concern families of $C^r$ sections of a singular set.
For real and complex analytic sets, we show that the $t^r$0 conditions
have algebraic formulations in terms of integral closure of modules.
Our formulation gives a new simple proof, for analytic sets, of the change
in the conditions under Grassmann modification proved by Kuo and Trotman for
subanalytic sets; this is used in conjuntion with the principle of specializtion
of integral dependence to give numerical criteria for families of plane sections
of complex complete intersections to be Whitney equisingular.
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