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Polynomials Inducing the Zero Function on Local Rings

Year 2017, , 170 - 186, 11.07.2017
https://doi.org/10.24330/ieja.325942

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 $\pi(x)$.

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.
Year 2017, , 170 - 186, 11.07.2017
https://doi.org/10.24330/ieja.325942

Abstract

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.
There are 13 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Mark W. Rogers This is me

Cameron Wickham This is me

Publication Date July 11, 2017
Published in Issue Year 2017

Cite

APA Rogers, M. W., & Wickham, C. (2017). Polynomials Inducing the Zero Function on Local Rings. International Electronic Journal of Algebra, 22(22), 170-186. https://doi.org/10.24330/ieja.325942
AMA Rogers MW, Wickham C. Polynomials Inducing the Zero Function on Local Rings. IEJA. July 2017;22(22):170-186. doi:10.24330/ieja.325942
Chicago Rogers, Mark W., and Cameron Wickham. “Polynomials Inducing the Zero Function on Local Rings”. International Electronic Journal of Algebra 22, no. 22 (July 2017): 170-86. https://doi.org/10.24330/ieja.325942.
EndNote Rogers MW, Wickham C (July 1, 2017) Polynomials Inducing the Zero Function on Local Rings. International Electronic Journal of Algebra 22 22 170–186.
IEEE M. W. Rogers and C. Wickham, “Polynomials Inducing the Zero Function on Local Rings”, IEJA, vol. 22, no. 22, pp. 170–186, 2017, doi: 10.24330/ieja.325942.
ISNAD Rogers, Mark W. - Wickham, Cameron. “Polynomials Inducing the Zero Function on Local Rings”. International Electronic Journal of Algebra 22/22 (July 2017), 170-186. https://doi.org/10.24330/ieja.325942.
JAMA Rogers MW, Wickham C. Polynomials Inducing the Zero Function on Local Rings. IEJA. 2017;22:170–186.
MLA Rogers, Mark W. and Cameron Wickham. “Polynomials Inducing the Zero Function on Local Rings”. International Electronic Journal of Algebra, vol. 22, no. 22, 2017, pp. 170-86, doi:10.24330/ieja.325942.
Vancouver Rogers MW, Wickham C. Polynomials Inducing the Zero Function on Local Rings. IEJA. 2017;22(22):170-86.