I Cut Down Mesquite Tree. Now What?

Q. I cut down a mesquite tree I thought might lift my walkway leading to my residence. I was worried about waterlines and roots. Will the roots die or will they continue to grow? 

Velvet mesquite, an Arizona native tree, can sucker when younger but many of the mesquite stop suckering from the base when they get older.
https://mswn.com/plants/prosopis-juliflora-arizona-native-mesquite/

A. If the mesquite tree was cut down so that its top was removed (the part of the trunk at the soil surface and a little bit below) the roots should die. I dont see any suckers from this tree when it gets older so it should not sucker once it is cut down.

There are some trees which grow from the roots after they have been cut down but most mesquite trees do not. There can be some genetic differences. Let it grow the following year, give it some water, and see if produces some suckers or not. Roots of most mesquite trees usually do not sucker and continue to live after the tree was cut down. If you see no new growth from the roots after the tree was cut down, then the entire tree is dead and the roots will slowly decompose in the soil in a few years. 

Kill Living Roots

 If a short stump remains after the tree was cut down, consider taking a 1 inch wood bit and an electric drill and drill vertical holes as deep as you can in the remaining stump. Drill them within an inch of the outside bark. Drill as many as you can but put them no further than 1 inch apart. It is not necessary to drill vertical holes in the center of the trunk because that wood is already dead. 

You dont need to drill holes in the center of the tree because that is already dead unless of course it is a palm tree.
https://www.familyhandyman.com/project/how-to-remove-a-tree-stump-painlessly/

The only living part of the trunk is close to the bark. Pour salt down these holes, copper sulfate or anything concentrated to kill the crown or use diluted weedkiller (look at the ingredients for glyphosate, dicamba, triclopyr) if the label permits it.  Some like salt and other homemade remedies are not systemic like glyphosate is and technically are not permitted.  Read the label and it will tell you what the dilution should be but it is usually the same dilution that is recommended for weed control.