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,…,x_n} over a field ${\mathbb {F}}$. An ideal I = 〈p₁(x₁),…,p_n(x_n)〉 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 p_i 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 ℓ_i : 1 ≤ i ≤ r are linear forms in X. We give a deterministic time d^O(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 n^O(r) algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields a new n^O(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 p_i has all distinct rational roots we can check if f ∈ I in randomized O^∗(n^k/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 $I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{k}^{e_{k}}}\right \rangle$.