Polynomials Inducing the Zero Function on Local Rings

Authors : Mark W ROGERS, Cameron WİCKHAM

Pages : 170-186

Doi:10.24330/ieja.325942

View : 12 | Download : 11

Publication Date : 2017-07-11

Article Type : Research Paper

Abstract :For a Noetherian local ring $insert ignore into journalissuearticles values(R, \f{m});$ having a finite residue field of   cardinality $q$, we study the connections between the ideal \zf{R} of $R[x]$,   which is the set of polynomials that vanish on $R$, and the ideal \zf{\f{m}},   the polynomials that vanish on \f{m}, using polynomials of the form   $\piinsert ignore into journalissuearticles values(x); = \prod_{i = 1}^{q} insert ignore into journalissuearticles values(x - c_{i});$, where $c_{1}, \ldots, c_{q}$ is a   set of representatives of the residue classes of \f{m}.  In particular, when   $R$ is Henselian we show that a generating set for \zf{R} may be obtained from   a generating set for \zf{\f{m}} by composing with $\piinsert ignore into journalissuearticles values(x);$.
Keywords : Finite ring, polynomial function