A computational model for algebraic power series

Encoding Algebraic Power Series

“…The mere existence follows from Hironaka's theorem. The effective part in the special case of principal ideals, i.e., the construction of the code of the Weierstrass form of an x n -regular algebraic power series, has been established by Alonso, Mora and Raimondo [AMR,Thm. 5.5].…”

“…Our main result asserts that the division by modules of algebraic power series vectors with box condition can be made effective, i.e., can be performed by applying finitely many operations to the codes. The case of principal ideals I, say the effective Weierstrass Division Theorem for algebraic power series, is due to Alonso, Mora and Raimondo in [AMR,Thm. 5.6].…”

“…Step 2: Specification of all x n -regular elements of this basis and computation of the reduced standard basis of the ideal I 1 generated by them. Here, x n is z; as only g 1 is z-regular, I 1 = g 1 is principal and its reduced standard basis can be computed with the algorithm of [AMR,Thm. 5.5] or, equivalently, as described in Theorem 10.1 above in the x n -regular case for principal ideals.…”

Elementary constructive theory of Henselian local rings

“…. ", this means that: (1) we know how to construct elements of R (from now on called canonical elements), (2) we have given 1 R , 0 R , −1 R , constructed according to (1), (3) we know how to construct x + y and xy according to (1), when the objects x and y are given through the same construction, (4) we know what is the meaning of x = R y when x and y are elements of R given through the construction (1), and (5) we have constructive proofs showing that the axioms of rings are satisfied by this structure.…”

“…This fact allows to represent algebraic functions (locally), and to state algorithms on standard bases in the ring of algebraic formal power series (cf. [1]). This characterization of the Henselization relies on Zariski Main Theorem, which provides a kind of "primitive smooth element" forétale extensions.…”