Computation of inverses in residue class rings of parametric polynomial ideals

  • Authors:

  • Yosuke Sato;Akira Suzuki

  • Affiliations:

  • Tokyo University of Science, Tokyo, Japan;Kobe University, Kobe, Japan

  • Venue:
  • Proceedings of the 2009 international symposium on Symbolic and algebraic computation
  • Year:
  • 2009

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Abstract

For a given polynomial f and an ideal I of a polynomial ring K[X] over a field K, we give a necessary and sufficient condition for I to have a smallest ideal extension J such that f is invertible in the residue class ring K[X]/J. If the condition holds, J is shown to be the saturation ideal I:∞. We also show primary decompositions of the ideals I:∞ and I+9fm;:, where m is a natural number such that I:∞ = I:fm, all together forms a primary decomposition of I with no modification. These observations are especially useful for polynomial rings with parameters.