# Polynomial decomposition

In mathematics, a **polynomial decomposition** expresses a polynomial *f* as the functional composition of polynomials *g* and *h*, where *g* and *h* have degree greater than 1.[1] Algorithms are known for decomposing polynomials in polynomial time.

Polynomials which are decomposable in this way are **composite polynomials**; those which are not are **prime** or **indecomposable polynomials**[2] (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials).

## Examples

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

decomposes into

since

Less trivially,

## Uniqueness

A polynomial may have distinct decompositions into indecomposable polynomials where where for some . 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 , and the degrees of the components are the same, but possibly in different order; this is **Ritt's polynomial decomposition theorem**.[2][3] For example, .

## Applications

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

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

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

the roots of this irreducible polynomial can be calculated as

- .[5]

Even in the case of quartic polynomials, where there is an explicit formula for the roots, solving using the decomposition often gives a simpler form. For example, the decomposition

gives the roots

but straightforward application of the quartic formula gives equivalent results but in a form that is difficult to simplify and difficult to understand:

## Algorithms

The first algorithm for polynomial decomposition was published in 1985,[6] though it had been discovered in 1976[7] and implemented in the Macsyma computer algebra system.[8] That algorithm took worst-case exponential time but worked independently of the characteristic of the underlying field.

More recent algorithms ran in polynomial time but with restrictions on the characteristic.[9]

The most recent algorithm calculates a decomposition in polynomial time and without restrictions on the characteristic.[10]

## Notes

- Composition of polynomials may also be thought of as substitution of one polynomial as the value of the variable of another.
- J.F. Ritt, "Prime and Composite Polynomials",
*Transactions of the American Mathematical Society***23**:1:51–66 (January, 1922) doi:10.2307/1988911 JSTOR 1988911 - Capi Corrales-Rodrigáñez, "A note on Ritt's theorem on decomposition of polynomials",
*Journal of Pure and Applied Algebra***68**:3:293–296 (6 December 1990) doi:10.1016/0022-4049(90)90086-W - The examples below were calculated using Maxima.
- Where each ± is taken independently.
- David R. Barton, Richard Zippel, "Polynomial Decomposition Algorithms",
*Journal of Symbolic Computation***1**:159–168 (1985) - Richard Zippel , "Functional Decomposition" (1996) full text
- Available in its open-source successor, Maxima, see the
*polydecomp*function - Dexter Kozen, Susan Landau, "Polynomial Decomposition Algorithms",
*Journal of Symbolic Computation***7**:445–456 (1989) - Raoul Blankertz, "A polynomial time algorithm for computing all minimal decompositions of a polynomial",
*ACM Communications in Computer Algebra***48**:1 (Issue 187, March 2014) full text Archived 2015-09-24 at the Wayback Machine

## References

- Joel S. Cohen, "Polynomial Decomposition", Chapter 5 of
*Computer Algebra and Symbolic Computation*, 2003, ISBN 1-56881-159-4