A polled animal is one that was born without horns. The polled trait is determined by a genetic variation on cattle chromosome 1.
Virtually all polled Dexters in the United States descend from a bull named Saltaire Platinum, (shown in this photo,) whose semen was imported from England by Fred Chesterley. The bull's granddam is believed to represent a fresh mutation. A fresh mutation is a term used to describe an animal who carries a genetic variation not found in it's parents. Platinum's grandmother had two horned parents, but had no horns. This is impossible except in the instance of a fresh mutation in the animal's DNA.
Buyers looking for polled cattle must be careful. Since polledness is a dominant trait, there are no "hidden" recessive polled genes lurking in horned animals. A horned animal out of one or even two polled parents does not carry the polled gene. Breeders who are not aware of this fact will market horned animals as carriers of the polled trait.
This is not to say that horned animals from the Platinum line are not desirable. As you can see from the picture, the Platinum line has more to offer than just the lack of horns. As well as a nice stocky build, the line also has nice udder characteristics.
The following represents the inheritance of polledness in cattle and the ratios of horned and polled animals you would get with various breedings. Letters are used in clinical genetics to represent alleles or different forms of a particular gene. This is a single gene trait, so each animal has two alleles - one from each parent. Since polled is dominant is gets the big P and since horns is recessive, it gets the little p. (See the page on genetics terms for more clarification.)