Polynomial transformation

In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.

Translating the roots
Let
 * $$ P(x) = a_0x^n + a_1 x^{n-1} + \cdots + a_{n} $$

be a polynomial, and
 * $$\alpha_1, \ldots, \alpha_n$$

be its complex roots (not necessarily distinct).

For any constant $c$, the polynomial whose roots are
 * $$\alpha_1+c, \ldots, \alpha_n+c$$

is
 * $$Q(y) = P(y-c)= a_0(y-c)^n + a_1 (y-c)^{n-1} + \cdots + a_{n}. $$

If the coefficients of $P$ are integers and the constant $$c=\frac{p}{q}$$ is a rational number, the coefficients of $Q$ may be not integers, but the polynomial $c^{n} Q$ has integer coefficients and has the same roots as $Q$.

A special case is when $$c=\frac{a_1}{na_0}.$$ The resulting polynomial $Q$ does not have any term in $y^{n &minus; 1}$.

Reciprocals of the roots
Let
 * $$ P(x) = a_0x^n + a_1 x^{n-1} + \cdots + a_{n} $$

be a polynomial. The polynomial whose roots are the reciprocals of the roots of $P$ as roots is its reciprocal polynomial
 * $$ Q(y)= y^nP\left(\frac{1}{y}\right)= a_ny^n + a_{n-1} y^{n-1} + \cdots + a_{0}.$$

Scaling the roots
Let
 * $$ P(x) = a_0x^n + a_1 x^{n-1} + \cdots + a_{n} $$

be a polynomial, and $c$ be a non-zero constant. A polynomial whose roots are the product by $c$ of the roots of $P$ is
 * $$Q(y)=c^nP\left(\frac{y}{c} \right) = a_0y^n + a_1 cy^{n-1} + \cdots + a_{n}c^n. $$

The factor $c^{n}$ appears here because, if $c$ and the coefficients of $P$ are integers or belong to some integral domain, the same is true for the coefficients of $Q$.

In the special case where $$c=a_0$$, all coefficients of $Q$ are multiple of $c$, and $$ \frac{Q}{c}$$ is a monic polynomial, whose coefficients belong to any integral domain containing $c$ and the coefficients of $P$. This polynomial transformation is often used to reduce questions on algebraic numbers to questions on algebraic integers.

Combining this with a translation of the roots by $$\frac{a_1}{na_0}$$, allows to reduce any question on the roots of a polynomial, such as root-finding, to a similar question on a simpler polynomial, which is monic and does not have a term of degree $n &minus; 1$. For examples of this, see Cubic function § Reduction to a depressed cubic or Quartic function § Converting to a depressed quartic.

Transformation by a rational function
All preceding examples are polynomial transformations by a rational function, also called Tschirnhaus transformations. Let
 * $$f(x)=\frac{g(x)}{h(x)}$$

be a rational function, where $g$ and $h$ are coprime polynomials. The polynomial transformation of a polynomial $P$ by $f$ is the polynomial $Q$ (defined up to the product by a non-zero constant) whose roots are the images by $f$ of the roots of $P$.

Such a polynomial transformation may be computed as a resultant. In fact, the roots of the desired polynomial $Q$ are exactly the complex numbers $y$ such that there is a complex number $x$ such that one has simultaneously (if the coefficients of $P, g$ and $h$ are not real or complex numbers, "complex number" has to be replaced by "element of an algebraically closed field containing the coefficients of the input polynomials")
 * $$\begin{align}

P(x)&=0\\ y\,h(x)-g(x)&=0\,. \end{align} $$ This is exactly the defining property of the resultant
 * $$\operatorname{Res}_x(y\,h(x)-g(x),P(x)).$$

This is generally difficult to compute by hand. However, as most computer algebra systems have a built-in function to compute resultants, it is straightforward to compute it with a computer.

Properties
If the polynomial $P$ is irreducible, then either the resulting polynomial $Q$ is irreducible, or it is a power of an irreducible polynomial. Let $$\alpha$$ be a root of $P$ and consider $L$, the field extension generated by $$\alpha$$. The former case means that $$f(\alpha)$$ is a primitive element of $L$, which has $Q$ as minimal polynomial. In the latter case, $$f(\alpha)$$ belongs to a subfield of $L$ and its minimal polynomial is the irreducible polynomial that has $Q$ as power.

Transformation for equation-solving
Polynomial transformations have been applied to the simplification of polynomial equations for solution, where possible, by radicals. Descartes introduced the transformation of a polynomial of degree $d$ which eliminates the term of degree $d − 1$ by a translation of the roots. Such a polynomial is termed depressed. This already suffices to solve the quadratic by square roots. In the case of the cubic, Tschirnhaus transformations replace the variable by a quadratic function, thereby making it possible to eliminate two terms, and so can be used to eliminate the linear term in a depressed cubic to achieve the solution of the cubic by a combination of square and cube roots. The Bring–Jerrard transformation, which is quartic in the variable, brings a quintic into Bring-Jerrard normal form with terms of degree 5,1, and 0.