Bounds on the number of steps of WDS required for checking the positivity of integral forms

Examples show that many integral forms can be efficiently verified to be positive through a special form of variable substitutions, i.e., weighted difference substitutions (WDS), but it was unknown how many steps of substitutions are needed, or furthermore, whether this method is adapted for checking positivity of all integral forms. In this paper, we give upper bounds of step numbers of WDS required in checking whether an integral form is positive or nonnegative, thus deducing that the positivity of integral forms can be completely verified through the WDS method.