Let be an algebraically closed field and . Let be an ideal. Since is noetherian, a theorem of Noether says that there are finitely many prime ideals such that .

Suppose that and are uniquely factored into irreducibles. Then each is equal to or . Suppose that of them are and of them are . Then

To have , therefore, we need

.

So we need or , i.e.

As vary, we may replace with . We therefore deduce:

**Bound 1. **.

**Bound 2.** .

**Remark.** Bound 1 is obviously stronger; however, since in general we don’t have information on and , bound 2 is uniformly the best possible.

The calculation becomes much more subtle for general polynomials.