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

Abstract

Let \({\mathbb {F}}[X]\) be the polynomial ring in the variables X = {x1,x2,…,xn} over a field \({\mathbb {F}}\). An ideal I = 〈p1(x1),…,pn(xn)〉 generated by univariate polynomials \(\{p_{i}(x_{i})\}_{i=1}^{n}\) is a univariate ideal. Motivated by Alon’s Combinatorial Nullstellensatz we study the complexity of univariate ideal membership: Given \(f\in {\mathbb {F}}[X]\) by a circuit and polynomials pi the problem is test if f ∈ I. We obtain the following results. Suppose f is a degree-d, rank-r polynomial given by an arithmetic circuit where ℓi : 1 ≤ i ≤ r are linear forms in X. We give a deterministic time dO(r) ⋅poly(n) division algorithm for evaluating the (unique) remainder polynomial f(X)modI at any point \(\vec {a}\in {\mathbb {F}}^{n}\). This yields a randomized nO(r) algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields a new nO(r) algorithm for evaluating the permanent of a n × n matrix of rank r, over any field \(\mathbb {F}\). Let f be over rationals with \(\deg (f)=k\) treated as fixed parameter. When the ideal \(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{n}^{e_{n}}}\right \rangle \), we can test ideal membership in randomized O∗((2e)k). On the other hand, if each pi has all distinct rational roots we can check if f ∈ I in randomized O∗(nk/2) time, improving on the brute-force \(\left (\begin {array}{cc}{n+k}\\ k \end {array}\right )\)-time search. If \(I=\left \langle {p_{1}(x_{1}), \ldots , p_{k}(x_{k})}\right \rangle \), with k as fixed parameter, then ideal membership testing is W[2]-hard. The problem is MINI[1]-hard in the special case when \(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{k}^{e_{k}}}\right \rangle \).