On Multivariate Rational Function Decomposition

On Multivariate Rational Function Decomposition

If K is a field, and g, h ∈ K(x) are rational functions of degree greater than one, then f = g ◦ h = g(h) is their (functional) composition, (g, h) is a (functional) decomposition of f , and f is a decomposable rational function. The univariate rational functional decomposition problem can be stated as follows: given f ∈ K(x), determine whether there exists a decomposition (g, h) of f with g an...

A recombination algorithm for the decomposition of multivariate rational functions

In this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study the complexity of this strategy and we show that this method improves the previous ones. In appendix, we explain how the strategy proposed recently by J. Berth...

On Multivariate Birkhoff Rational Interpolation

Decomposition of multivariate function using the Heaviside step function

Whereas the Dirac delta function introduced by P. A. M. Dirac in 1930 to develop his theory of quantum mechanics has been well studied, a not famous formula related to the delta function using the Heaviside step function in a single-variable form, also given by Dirac, has been poorly studied. Following Dirac's method, we demonstrate the decomposition of a multivariate function into a sum of int...

A note on commutative multivariate rational series

Direct arguments are presented showing that for rational series in several com muting variables the rational series problem is non computable and closure under Hadamard product fails