I have a garden plot that has been here for years, and when we moved in there was an abundant amount of "pigweed". I have tried burning them, spraying them, and continuous tilling. How do I completely get rid of them out of my garden?
Pigweed is pernicious, as you well know. A single plant can produce hundreds of thousands of seeds in a season. Seeds can hang around for at least 4 years before germinating. Burning will eliminate the current plants but opens the soil for seed germination. Some pigweeds have become herbicide resistant. A heavy mulch (mulched leaves would be good) should help keep seeds from germinating while it improves your soil. Continue the tilling, pull out flowering plants and dispose of them properly, i.e. do not put in your compost pile. Do not allow any pigweed within range of your garden to produce seed. All of that should help control pigweed. Getting rid of it completely is difficult. Birds eat pigweed seeds which means that birds spread pigweed seeds.
Best of luck.