In many cases using a multiplication table (ie, computing how a variable acts on residue fields of points in two different ways) can be used to more quickly decompose ideals. This is turned on by setting DecompositionStrategy => MultiplicationTable. However, in other cases, especially when there are many variables, using DecompositionStrategy => Decompose can be substantially faster. Currently MultiplicationTable only works with homogeneous ideals and will not find geometric points.
The object DecompositionStrategy is a symbol.