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.

Simple examples

Translating the roots

Let

P(x)=a_{0}x^{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 $y n - 1$ .

Reciprocals of the roots

Let

P(x)=a_{0}x^{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^{n}P\left({\frac {1}{y}}\right)=a_{n}y^{n}+a_{n-1}y^{n-1}+\cdots +a_{0}.

Scaling the roots

Let

P(x)=a_{0}x^{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^{n}P\left({\frac {y}{c}}\right)=a_{0}y^{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 $n - 1$ . 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{aligned}P(x)&=0\\y\,h(x)-g(x)&=0\,.\end{aligned}}

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.

References

Adamchik, Victor S.; Jeffrey, David J. (2003). "Polynomial transformations of Tschirnhaus, Bring and Jerrard" (PDF). SIGSAM Bull. 37 (3): 90–94. Zbl 1055.65063. Archived from the original (PDF) on 2009-02-26.