The ideal membership problem and polynomial identity testing

AbstractGiven a monomial ideal I=〈m1,m2,…,mk〉 where mi 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=〈m1,m2,…,mk〉 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=〈m1,m2,…,mk〉 for a constant kandf is computed by a ΣΠΣ 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 ΣΠΣ 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 coMAPP over the field of rationals, and by coNPModpP over finite fields.(d)Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits.For ideals I=〈f1,…,fℓ〉 where each fi∈F[x1,…,xk] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above

Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of G\"artner et al.\ applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange

Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve

Efficient Black-Box Identity Testing for Free Group Algebras

Hrubes and Wigderson [Pavel Hrubes and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n

Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Let F[X] be the polynomial ring over the variables X={x_1,x_2, ..., x_n}. An ideal I= generated by univariate polynomials {p_i(x_i)}_{i=1}^n is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results. - Let f(X) in F[l_1, ..., l_r] be a (low rank) polynomial given by an arithmetic circuit where l_i : 1 be a univariate ideal. Given alpha in F^n, the (unique) remainder f(X) mod I can be evaluated at alpha in deterministic time d^{O(r)} * poly(n), where d=max {deg(f),deg(p_1)...,deg(p_n)}. This yields a randomized n^{O(r)} algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n^{O(r)} algorithm for evaluating the permanent of a n x n matrix of rank r, over any field F. Over Q, an algorithm of similar run time for low rank permanent is due to Barvinok [Barvinok, 1996] via a different technique. - Let f(X)in F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I=. We show that in the special case when I=, we obtain a randomized O^*(4.08^k) algorithm that uses poly(n,k) space. - Given f(X)in F[X] by an arithmetic circuit and I=, membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I=