Decomposition of polynomials and approximate roots
Arnaud Bodin · Proceedings of the American Mathematical Society · 2010
We state a kind of Euclidian division theorem: given a polynomial P ( x ) P(x) and a divisor d d of the degree of P P , there exist polynomials h ( x ) , Q ( x ) , R ( x ) h(x),Q(x),R(x) such that P ( x ) = h ∘ Q ( x ) + R ( x ) P(x) = h\circ Q(x) +R(x) , with deg h = d \deg h=d . Under some conditions h , Q , R h,Q,R are unique, and Q Q is the approximate d d -root of P P . Moreover we give an algorithm to compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not.