I read somewhere that polygel needs a lamp with a minimum of 48 watts. IDK if that's true... I'm not a nail tech! But those mini lamps are around 6w. You can get an LED lamp with 48w or more for less than $20 on Amazon.
As far as brands, Makartt, Modelones, Beetles, and Morovan are all reasonably priced and highly rated. You can even buy kits that come with a 48w lamp for around $40! Examples of sets with lamps -