They are not bamboo. These are types of Smilax sp.. There are many different forms - some of them have the potato type tubers and some have elaborate "knotty" roots.

Yes, digging them up and disposing of the tuber will absolutely work. Cutting them down year after year (it does take several years for them to grow so tall into the trees) can help deprive them of the energy to keep growing. If you cut them and then apply a brush killer (like Brush-B-Gon) to the stump, that may help as well.

The ones in your flower beds are probably 1 or 2 year seedlings.

Here is a link that might be useful: Here is just one of the common ones

Esh has good advice and advised me well on getting rid of our smilax vine. Seedlings will continue to germinate over time so make sure you take a look around every year to get them while they are young. I have gotten out all the bigger/older vines successfully now and just fight the battle of a few smaller vines each year...

GGG

Smilax sp. are a favorite food source of birds, so you will never be free of the vines. There are 20 Smilax species in the US and 14 are native to GA. 9 of those are found state-wide or nearly so and you may find several different ones in your garden.
I have found one that is an attractive plant and I grow it as an ornamental. Smilax pumila(humilis) has red berries, bronze winter foliage and no thorns, a few prickles are found sometimes near the base and has no tendrils(non-climbing). Usually grows upright for only a few feet or trails on the ground.
Smilax smallii is the highest climbing species, reaching heights of more than 30 feet. It has a few prickles, but no thorns. Smilax laurifolia is the most viciously armed of the species.
Good luck with eradicating them! I usually find one or two each year that has escaped detection among large Azaleas and the battle is on!

Yes,Old Man, you are right. It will come up everywhere you don't want it. If you leave even a smidgen of root or rhizome in the dirt, it will shoot right back up. When we moved to our new house, near a creek bank, we dug our guts out trying to get rid of it, and sprayed with Brush-B-Gone as well. It has been 9 years and we still dig up Similax, but there are fewer each year. It especially likes to come up in the middle of shrubs where it is almost impossible to dig it out.

You can stick a piece of vine in a jar of roundup and let it sit/absorb a few days. This will get rid of difficult vines easily.