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

${\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}}$

be a polynomial, and

${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$

be its complex roots (not necessarily distinct).

For any constant c, the polynomial whose roots are

${\displaystyle \alpha _{1}+c,\ldots ,\alpha _{n}+c}$

is

${\displaystyle Q(y)=P(y-c)=a_{0}(y-c)^{n}+a_{1}(y-c)^{n-1}+\cdots +a_{n}.}$

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

A special case is when ${\displaystyle c={\frac {a_{1}}{na_{0}}}.}$ The resulting polynomial Q does not have any term in y^{n− 1}.

Reciprocals of the roots

Let

${\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}}$

be a polynomial. The polynomial whose roots are the reciprocals of the roots of P as roots is its reciprocal polynomial

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

Scaling the roots

Let

${\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}}$

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

${\displaystyle Q(y)=c^{n}P\left({\frac {y}{c}}\right)=a_{0}y^{n}+a_{1}cy^{n-1}+\cdots +a_{n}c^{n}.}$

The factor cⁿ 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 ${\displaystyle c=a_{0}}$, all coefficients of Q are multiple of c, and ${\displaystyle {\frac {Q}{c}}}$ 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 ${\displaystyle {\frac {a_{1}}{na_{0}}}}$, 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

${\displaystyle f(x)={\frac {g(x)}{h(x)}}}$

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")

${\displaystyle {\begin{aligned}P(x)&=0\\y\,h(x)-g(x)&=0\,.\end{aligned}}}$

This is exactly the defining property of the resultant

${\displaystyle \operatorname {Res} _{x}(y\,h(x)-g(x),P(x)).}$

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 ${\displaystyle \alpha }$ be a root of P and consider L, the field extension generated by ${\displaystyle \alpha }$. The former case means that ${\displaystyle f(\alpha )}$ is a primitive element of L, which has Q as minimal polynomial. In the latter case, ${\displaystyle f(\alpha )}$ 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.