The ideal membership problem and polynomial identity testing
نویسندگان
چکیده
منابع مشابه
The ideal membership problem and polynomial identity testing
Given a monomial ideal I = 〈m1,m2, · · · ,mk〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I . We study this problem and show the following results. (a) If the ideal I = 〈m1,m2, · · · ,mk〉 for a constant k then there is a randomized polynomial-time membership algorithm to test if f ∈ I . This result holds even for f given by a...
متن کاملOn the Parallel Complexity of the Polynomial Ideal Membership Problem
The complexity of the polynomial ideal membership problem over arbitrary fields within the framework of arithmetic networks is investigated. We prove that the parallel complexity of this problem is single exponential over any infinite field. Our lower bound is obtained by combining a modification of Mayr and Meyer's (1982) key construction with an elementary degree bound. 1998 Academic Press, Inc.
متن کاملIdeal Membership Problem
The Ideal Membership Problem is as follows: given f0, f1, . . . , fm ∈ K[x1, . . . , xn], is f0 ∈ 〈f1, . . . , fm〉, where 〈f1, . . . , fm〉 denotes the ideal generated by the fi? An equivalent formulation is: are there q1, . . . , qm ∈ K[x1, . . . , xn] such that f0 = ∑m i=1 qifi? We will solve this question by using Gröbner bases. That is, a Gröbner basis is a “nice” representation of an ideal,...
متن کاملPolynomial Identity Testing
holds. A natural interpretation could be the following: Definition for interpretation 1: p(x1, x2, . . . , xn) ≡ 0 if for each a1, a2, . . . , an ∈ D, p(a1, a2, . . . , an) = 0 where D denotes the domain in which the polynomial is defined. However, there’s another interpretation: Definition for interpretation 2: p(x1, x2, . . . , xn) ≡ 0 if in the form where the polynomial is written as the lin...
متن کاملIdeal Membership in Polynomial Rings over the Integers
We will reproduce a proof, using Hermann’s classical method, in Section 3 below. Note that the computable character of this bound reduces the question of whether f0 ∈ (f1, . . . , fn) for given fj ∈ F [X ] to solving an (enormous) system of linear equations over F . Hence, in this way one obtains a (naive) algorithm for solving the ideal membership problem for F [X ] (provided F is given in som...
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ژورنال
عنوان ژورنال: Information and Computation
سال: 2010
ISSN: 0890-5401
DOI: 10.1016/j.ic.2009.06.003