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 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 called indecomposable polynomials of sometimes prime polynomials. (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials).
The rest of this article discusses only univariate polynomials; algorithms also exist for multivariate polynomials of arbitrary degree.
In the simplest case, one of the polynomials is a monomial. For example,
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.
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
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.
- 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
- Jean-Charles Faugère, Ludovic Perret, "An efficient algorithm for decomposing multivariate polynomials and its applications to cryptography", Journal of Symbolic Computation, 44:1676-1689 (2009), doi:10.1016/j.jsc.2008.02.005
- 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