Polynomial decomposition

In mathematics, a polynomial decomposition expresses a polynomial f as the functional composition ${\displaystyle g\circ h}$ of polynomials g and h, where g and h have degree greater than 1; it is an algebraic functional decomposition. Algorithms are known for decomposing univariate polynomials in polynomial time.

Polynomials which are decomposable in this way are composite polynomials; those which are not are indecomposable polynomials or sometimes prime polynomials¹ (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials). The degree of a composite polynomial is always a composite number, the product of the degrees of the composed polynomials.

The rest of this article discusses only univariate polynomials; algorithms also exist for multivariate polynomials of arbitrary degree.²³⁴

In the simplest case, one of the polynomials is a monomial. For example,

${\displaystyle f(x)=x^{6}-3x^{3}+1}$

decomposes into ${\displaystyle g\circ h}$, where

${\displaystyle g(x)=x^{2}-3x+1{\text{ and }}h(x)=x^{3}}$

since

${\displaystyle f(x)=(g\circ h)(x)=g(h(x))=g(x^{3})=(x^{3})^{2}-3(x^{3})+1,}$

using the ring operator symbol ${\displaystyle {\boldsymbol {\circ }}}$ to denote function composition. We write that as

${\displaystyle x^{6}-3x^{3}+1=(x^{2}-3x+1)\circ (x^{3})}$

treating the polynomials implicitly as functions of ${\displaystyle x}$.

Less trivially,

${\displaystyle {\begin{aligned}&x^{6}-6x^{5}+21x^{4}-44x^{3}+68x^{2}-64x+41\\={}&(x^{3}+9x^{2}+32x+41)\circ (x^{2}-2x).\end{aligned}}}$

A polynomial may have distinct decompositions into indecomposable polynomials where ${\displaystyle f=g_{1}\circ g_{2}\circ \cdots \circ g_{m}=h_{1}\circ h_{2}\circ \cdots \circ h_{n}}$ where ${\displaystyle g_{i}\neq h_{i}}$ for some ${\displaystyle i}$. The restriction in the definition to polynomials of degree greater than one excludes the infinitely many decompositions possible with linear polynomials.

Joseph Ritt proved that ${\displaystyle m=n}$, and the degrees of the components are the same, but possibly in different order; this is Ritt's polynomial decomposition theorem.¹⁵ For example, ${\displaystyle x^{2}\circ x^{3}=x^{3}\circ x^{2}}$. In fact, only three kinds of position swap are possible: between monomials; between Chebyshev polynomials ${\displaystyle T_{i}(x)}$; and between ${\displaystyle \left(x^{k}\,u\left(x\right)^{p},x^{p}\right)\leftrightarrow \left(x^{p},x^{k}\,u\left(x^{p}\right)\right)}$— summarized as "Every polynomial can be written as a composition of indecomposables, uniquely up to permutations and units."⁶ For example:

${\displaystyle x^{14}\,\left(x^{98}+1\right)^{2}=\left(x\,\left(x^{7}+1\right)^{2}\right)\circ \left(x^{7}\right)\circ \left(x^{2}\right)=\left(x^{2}\right)\circ \left(x\,\left(x^{14}+1\right)\right)\circ \left(x^{7}\right)}$

${\displaystyle T_{2}(x)\circ T_{3}(x)=T_{3}(x)\circ T_{2}(x)=32\,x^{6}-48\,x^{4}+18\,x^{2}-1}$

A polynomial decomposition may enable more efficient evaluation of a polynomial. For example,

${\displaystyle {\begin{aligned}&x^{8}+4x^{7}+10x^{6}+16x^{5}+19x^{4}+16x^{3}+10x^{2}+4x-1\\={}&\left(x^{2}-2\right)\circ \left(x^{2}\right)\circ \left(x^{2}+x+1\right)\end{aligned}}}$

can be calculated with 3 multiplications and 3 additions using the decomposition, while Horner's method would require 7 multiplications and 8 additions.

A polynomial decomposition enables calculation of symbolic roots using radicals, even for some irreducible polynomials of degree greater than 4. This technique is used in many computer algebra systems.⁷ For example, using the decomposition

${\displaystyle {\begin{aligned}&x^{6}-6x^{5}+15x^{4}-20x^{3}+15x^{2}-6x-1\\={}&\left(x^{3}-2\right)\circ \left(x^{2}-2x+1\right),\end{aligned}}}$

the roots of this irreducible polynomial can be calculated as⁸

${\displaystyle 1\pm 2^{1/6},1\pm {\frac {\sqrt {-1\pm {\sqrt {3}}i}}{2^{1/3}}}.}$

In the case of quartic polynomials, if there is a decomposition, it can give a simpler form than the general formula. For example, the decomposition

${\displaystyle {\begin{aligned}&x^{4}-8x^{3}+18x^{2}-8x+2\\={}&(x^{2}+1)\circ (x^{2}-4x+1)\end{aligned}}}$

gives the roots⁸

${\displaystyle 2\pm {\sqrt {3\pm i}}}$

but straightforward application of the quartic formula gives a form that is difficult to simplify and difficult to understand; one of the four roots is:

${\displaystyle 2-{\frac {\sqrt {{9\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{2/3}+36\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}+156} \over {\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}}}}{6}}-{{\sqrt {-\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}-{{52} \over {3\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}}}+8}} \over 2}.}$

The first algorithm for polynomial decomposition was published in 1985,⁹ though it had been discovered in 1976,¹⁰ and implemented in the Macsyma/Maxima computer algebra system.¹¹ That algorithm takes exponential time in worst case, but works independently of the characteristic of the underlying field.

A 1989 algorithm runs in polynomial time but with restrictions on the characteristic.¹²

A 2014 algorithm calculates a decomposition in polynomial time and without restrictions on the characteristic.¹³