The ideal membership problem and polynomial identity testing

  • Authors:

  • V. Arvind;Partha Mukhopadhyay

  • Affiliations:

  • Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600 113, India;Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600 113, India

  • Venue:
  • Information and Computation
  • Year:
  • 2010

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Abstract

Given a monomial ideal I= where m"i are monomials and a polynomial f by an arithmetic circuit, the Ideal Membership Problem is to test if f@?I. We study this problem and show the following results. (a)When the ideal I= for a constant k, we can test whether f@?I in randomized polynomial time. This result holds even for f given by a black-box, when f is of small degree. (b)When I= for a constantkandf is computed by a @S@P@S circuit with output gate of bounded fanin, we can test whether f@?I in deterministic polynomial time. This generalizes the Kayal-Saxena result [11] of deterministic polynomial-time identity testing for @S@P@S circuits with bounded fanin output gate. (c)When k is not constant the problem is coNP-hard. We also show that the problem is upper bounded by coMA^P^P over the field of rationals, and by coNP^M^o^d^p^P over finite fields. (d)Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits. For ideals I= where each f"i@?F[x"1,...,x"k] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above.