On Multivariate Rational Function Decomposition

Abstract

In this paper we discuss several notions of decomposition for multivariate rational functions, and we present algorithms for decomposing multivariate rational functions over an arbitrary field. We also provide a very efficient method to decide if a unirational field has transcendence degree one, and in the affirmative case to compute the generator.

On Ritt's decomposition theorem in the case of finite fields

“…See for example [9] for a proof in the case K = C, [15] for one in the general case and [16] for an elementary one. Constructive proofs can be found in [10,13,1]. Now, we state one of the classical Ritt's theorems (see [11]) about the relations among the complete decompositions of a polynomial that satisfies a certain condition.…”

Implicitization using univariate resultants

“…The decomposition method for rational functions [15] will be used and more specifically, we only concern about uni-multivariate decomposition.…”

“…That is, min{deg s (h), deg t (h)} = 1 and we call such h as quasi-linear. Fortunately, we can use the uni-multivariate rational decomposition method [15] to solve this problem.…”

“…We modify the algorithm 2.1 in [15] to find quasi-linear decomposition factors. Algorithm 1 Input: f = fn(s,t) f d (s,t) ∈ C(s, t).…”

Common composites of triangular polynomial systems and hash functions

“…For multivariate polynomials and rational functions, it is not even clear what the most interesting notions of decompositions are, see Perret (2009a, 2009b), Faugère et al (2010), von zur Gathen et al (2003), Gutierrez et al (2002) for different approaches, as well as connections to intermediate subfield problems, which are extensions of the univariate case, and applications to cryptography.…”