A polynomial time algorithm for computing all minimal decompositions of a polynomial

Abstract

The composition of two polynomials g(h) = g • h is a polynomial. For a given polynomial f we are interested in finding a functional decomposition f = g • h. In this paper an algorithm is described, which computes all minimal decompositions in polynomial time. In contrast to many previous decomposition algorithms this algorithm works without restrictions on the degree of the polynomial and the characteristic of the ground field. The algorithm can be iteratively applied to compute all decompositions. It is based on ideas of Landau & Miller (1985) and Zippel (1991). Additionally, an upper bound on the number of minimal decompositions is given.

Computer Algebra and Polynomials

“…Zippel (1991) suggests that the block decompositions of Landau and Miller (1985) for determining subfields of algebraic number fields can be applied to decomposing rational functions even in the wild case. A version of Zippel's algorithm in Blankertz (2014) computes in polynomial time all decompositions of a polynomial that are minimal in a certain sense. Avanzi and Zannier (2003) study ambiguities in the decomposition of rational functions over C. On a different but related topic, Zieve and Müller (2008) found interesting characterizations for Ritt's First Theorem, which deals with complete decompositions, where all components are indecomposable.…”

Survey on Counting Special Types of Polynomials

“…Zippel (1991) suggests that the block decompositions of Landau & Miller (1985) for determining subfields of algebraic number fields can be applied to decomposing rational functions even in the wild case. A version of Zippel's algorithm in Blankertz (2014) computes in polynomial time all decompositions of a polynomial that are minimal in a certain sense. Avanzi & Zannier (2003) study ambiguities in the decomposition of rational functions over C. On a different but related topic, Zieve & Müller (2008) found interesting characterizations for Ritt's First Theorem, which deals with complete decompositions, where all components are indecomposable.…”

Functional Decomposition Using Principal Subfields

“…Further improvements are presented in [20,21]. More recently, [7] presented a polynomial time algorithm that finds all minimal decompositions of f , with no restrictions on deg( ) or the characteristic of the field.…”

“…When char(K) > 0, the factorization of f (x) − f (t) can be computed with Õ(n ω+1 ) field operations, where 2 < ω ≤ 3 is a matrix multiplication exponent (see [8] and [14]). An algorithm in [7] also computes all minimal decompositions, and take Õ(n 6 ) field operations (for finite fields). For more details, see [6, eorem 3.23].…”