Polynomials Inducing the Zero Function on Local Rings

Year 2017, Volume: 22 Issue: 22, 170 - 186, 11.07.2017

Mark W. Rogers Cameron Wickham

https://doi.org/10.24330/ieja.325942

Cited By: 5

Abstract

For a Noetherian local ring $(R, \f{m})$ 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
  $\pi(x) = \prod_{i = 1}^{q} (x - c_{i})$, 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 $\pi(x)$.

Keywords

Finite ring, polynomial function

References

• A. Bandini, Functions f : Z{pnZ ÝÑ Z{pnZ induced by polynomials of Zrxs, Ann. Mat. Pura Appl., 181 (2002), 95-104.
• L. E. Dickson, Introduction to the Theory of Numbers, The University of Chicago Press, 1929.
• S. Frisch, Polynomial functions on finite commutative rings, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, (1999), 323-336.
• R. Gilmer, The ideal of polynomials vanishing on a commutative ring, Proc. Amer. Math. Soc., 127(5) (1999), 1265-1267.
• J. J. Jiang, On the number counting of polynomial functions, J. Math. Res. Exposition, 30(2) (2010), 241-248.
• J. Lahtonen, J. Ryu and E. Suvitie, On the degree of the inverse of quadratic permutation polynomial interleavers, IEEE Trans. Inform. Theory, 58(6) (2012), 3925-3932.
• D. J. Lewis, Ideals and polynomial functions, Amer. J. Math., 78 (1956), 71-77.
• B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics, 28, Marcel Dekker, Inc., New York, 1974.
• I. Niven and L. J. Warren, A generalization of Fermat's theorem, Proc. Amer. Math. Soc., 8 (1957), 306-313.
• G. Peruginelli, Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power, J. Algebra, 398 (2014), 227-242.
• N. J. Werner, Polynomials that kill each element of a finite ring, J. Algebra Appl., 13(3) (2014), 1350111 (12 pp).
• O. Zariski and P. Samuel, Commutative Algebra, vol. I, Springer-Verlag, Berlin-Heidelberg, 1958.
• Q. Zhang, Polynomial functions and permutation polynomials over some finite commutative rings, J. Number Theory, 105(1) (2004), 192-202.