Effective power series computations (Q2420635)

Let \(K\) be a field of characteristic zero and \[K[[z_1, z_2, \ldots]] = K \cup K[[z_1]] \cup K[[z_1, z_2]] \cup \cdots .\] Each element \(f \in K[[z_1, z_2, \ldots]]\) is a power series in a finite number of variables. \(K\) is called effective if its elements can be represented by concrete data structures and all field operations can be carried out by algorithms. Moreover, \(K\) admits an effective zero test if we also have an algorithm which takes \(f \in K\) as input and returns true if \(f = 0\) and false otherwise. Let \(L\) be an effective tribe (that is, a subset of \(K[[z_1, z_2, \ldots]]\) which is effectively stable under the \(K\)-algebra operations, restricted division, composition, the implicit function theorem and restricted monomial transformations with arbitrary rational exponents) over \(K\) with an effective zero test. Assume that \(f\in L\cap K[[z_1, \ldots, z_n]]\) has Weierstrass degree \(d\) in \(z_1\) (that is, \(f (0) = (\partial f /\partial z_1)(0) = \cdots = (\partial^{d-1} f / \partial z_1^{d-1})(0) = 0\), but \((\partial^{d} f / \partial z_1^{d} )(0) \ne 0)\) and let \(g \in L\cap K[[z_1, \ldots, z_n]]\). The Weierstrass division theorem states that there exist unique \(Q \in K[[z_1, \ldots , z_n]]\) and \(R \in K[[z_2, \ldots , z_n]][z_1]\) with \(g = Q f + R\) and \(\deg_{z_1}( R) < d\). Using these notions, the author introduces different notions including Weierstrass bases, Hironaka division and standard bases.