# How to depress a general polynomial

This post showed how to do a change of variables to remove the quadratic term from a cubic equation. Here we will show that the technique works more generally to remove the xn-1 term from an nth degree polynomial.

We will use big-O notation O(xk) to mean terms involving x to powers no higher than k. This is slightly unusual, because typically big-O notation is used when some variable is tending to a limit, and we’re not taking limits here. Here a is not zero, or else we wouldn’t have an nth degree polynomial.
The following calculation shows that the change of variables results in an nth degree polynomial in t with no term involving xn – 1. ## Finite fields

This approach works over real or complex numbers. It even works over finite fields too, if you can divide by na.

I’ve mentioned a couple times that the Weierstrass form of an elliptic curve is the most general except when working over a field of characteristic 2 or 3. The technique above breaks down because 3a may not be invertible in a field of characteristic 2 or 3.

## One thought on “How to depress a general polynomial”

1. What properties does the depressed polynomial share with the original? Are they exactly identical? Surely they must differ in some ways, and are they ever interesting ways?