Definition:Depressed Polynomial

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Definition

Let $f \left({x}\right)$ be a polynomial over a field $k$:

$f \left({x}\right) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0$

If $a_{n-1} = 0_k$, then we call $f$ a depressed polynomial.

It has been suggested that a polynomial with further zero terms might be referred to as "downright despondent", though this convention has yet to gain widespread usage by the community.

Tschirnhaus Substitution

When looking for solutions $f \left({x}\right) = 0$, we can make the linear substitution $x = y - \frac {a_{n-1}} n$.

Letting $O(y^j)$, $j \in \Z$ formally denote any finite sum of terms of degree at most $j$, we find that

\(\displaystyle a_n x^n + a_{n-1} x^{n-1} + O(x^{n-2})\) \(=\) \(\displaystyle a_n \left( y - \frac {a_{n-1} } n \right)^n + a_{n-1} \left( y - \frac {a_{n-1} } n \right)^{n-1} + O(y^{n-2})\)
\(\displaystyle \) \(=\) \(\displaystyle a_n \sum_{i = 0}^n {n \choose i} y^{n-i}\left(- \frac {a_{n-1} } n\right)^i + a_{n-1}\sum_{i = 0}^{n-1} {n-1 \choose i} y^{n-1-i}\left(- \frac {a_{n-1} } n\right)^i + O(y^2)\) using the Binomial Theorem
\(\displaystyle \) \(=\) \(\displaystyle a_n \left( y^n - ny^{n-1}\frac {a_{n-1} } n \right) + a_{n-1} y^{n-1} + O(y^2)\)
\(\displaystyle \) \(=\) \(\displaystyle a_n y^n + O(y^2)\)

This shows that the search for roots of $f$ can be reduced to the case when $f$ is depressed.

This technique is used in the derivation of Cardano's Formula for the roots of the general cubic.

It is known as a Tschirnhaus Transformation.