Ask Mr. Smarty Plants

Sapindus saponaria var. drummondii (Western soapberry)  often suckers and forms groves. Tolerant of drought, wind, heat, poor soil, air pollution and other city conditions. Not affected by disease or insects. Currently difficult to find in the nursery trade.

We really don't know if your sprouts are the product of berries that have lain in the soil since you cut the tree down, but we're betting that most, if not all, are adventitious sprouts from the still-living roots beneath the ground. As far as those roots are concerned, that tree is still alive. It puts out those sprouts not just to propagate itself into a grove, but to grow leaves to provide nutrition to the roots. Theoretically, if you cut off the sprouts long enough, the roots would eventually starve. Theoretically. Spraying herbicides will kill plants you didn't want killed, but won't affect the roots. We would recommend close mowing, if the sprouts are in grass. If you can find what we call a lopping shear at your hardware store, with heavy duty pruning blades and a handle long enough that you don't have to stoop, you can cut those sprouts as far down in the ground as you can get. If you encounter a root that you can cut, you could then try painting the cut surface with an undiluted wide spectrum herbicide, quickly, before the root heals over to protect itself. A disposable sponge brush is the best for this operation. Patience and persistence is the key. Don't ever let those "baby" trees get big enough to produce seed, that will just double your trouble.