We study a deformation of the Kummer sequence to the radicial sequence over an -algebra, which is somewhat dual for the deformation of the Artin-Schreier sequence to the radicial sequence, studied by Saidi. We also discuss some relations between our sequences and the embedding of a finite flat commutative group scheme into a connected smooth affine commutative group schemes, constructed by Grothendieck.
This note summarizes a presentation made at the Third International Meeting on Integer Valued Polynomials and Problems in Commutative Algebra. All the work behind it is joint with Scott T. Chapman, and will appear in [2]. Let represent the ring of polynomials with rational coefficients which are integer-valued at integers. We determine criteria for two such polynomials to have the same image set on .
Let be polynomials in variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials such that . The effective versions of this result bound the degrees of the in terms of the degrees of the . The aim of this paper is to generalize this to the case when the are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.