Ask Mr. Smarty Plants

One widely-used method is to either cut down the plant near ground level or girdle the trunk with an axe or chainsaw so that the cambium layer is exposed.  The cambium layer is the ring of cells just beneath the bark.  It's functions include transporting nutrients made in the leaves by photosynthesis down into the stem and roots and transporting water from the roots up to the leaves. Some woody plants can be killed simply by cutting through the cambium, but other species, including Nandina and Ligustrum, can sprout new growth from the stump. To prevent this the cut cambium layer should be painted with CONCENTRATED Roundup herbicide (not the diluted form sold to spray on leafy vegetation).  The herbicide should be applied within a very few minutes after disrupting the cambium.  Otherwise it will not be taken up by the plant effectively. 

Roundup concentrate can be obtained from some garden centers or hardware stores.  Roundup persists in nature for only a few days, but it is toxic to some animals while it lasts.  That is why you must carefully paint it on with a small brush rather than spraying over a larger area.  And do not use it withing a few feet of a stream or pond because rain could carry it into the water, where it would kill aquatic organisms.

Although this kind of  herbicide treatment will usually kill the main plant stem, certain invasive plants, such as Tree of heaven, are capable of sending up new shoots from roots some distance away from the killed parent plant. That is not likely to happen with Nandina or Wax-leaf ligustrum.  However, it is wise to monitor the situation for some months to assure that new suckers are not appearing in the vicinity.