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; it is a purely algebraic functional decomposition. 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.

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. 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. For example, using the decomposition
the roots of this irreducible polynomial can be calculated as
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, though it had been discovered in 1976 and implemented in the Macsyma computer algebra system. That algorithm takes worst-case exponential time but works independently of the characteristic of the underlying field.
More recent algorithms run in polynomial time but with restrictions on the characteristic.
The most recent algorithm calculates a decomposition in polynomial time and without restrictions on the characteristic.