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 \(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{k}^{e_{k}}}\right \rangle \).

令\({\mathbb {F}}[X]\)是变量X = { x ₁ , x ₂ ,..., x _n } 中的多项式环在域\({\mathbb {F}}\) . 由单变量多项式\(\{p_{i}(x_{i})\}_{i=1}^{n生成的理想I = 〈p ₁ ( x ₁ ),…, p _n ( x _n )〉}\)是一个单变量的理想。受 Alon 的组合 Nullstellensatz 的启发，我们研究了单变量理想隶属度的复杂性：给定\(f\in {\mathbb {F}}[X]\)通过电路和多项式p _i问题是测试f ∈ I。我们得到以下结果。

• 假设f是由算术电路给出的d阶、r阶多项式，其中ℓ _i : 1 ≤ i ≤ r是X中的线性形式。我们给出了一个确定性时间d ^{O ( r )} ⋅poly( n ) 除法算法，用于在任何点评估（唯一的）余数多项式f ( X )mod I \(\vec {a}\in {\mathbb {F}} ^{n}\)。这产生了一个随机的n ^{O ( r )}具有秩r邻接矩阵的图中最小顶点覆盖的算法。它还产生了一个新的n ^{O ( r )}算法，用于在任何字段\(\mathbb {F}\) 上评估n × n秩为r 的矩阵的永久值。

• 令f超过有理数，\(\deg (f)=k\)被视为固定参数。当理想\(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{n}^{e_{n}}}\right \rangle \)，我们可以测试理想随机O ^∗ ((2 e ) ^k ) 中的成员资格。另一方面，如果每个p _i都有所有不同的有理根，我们可以在随机O ^∗ ( n ^{k /2} ) 时间内检查f ∈ I是否改进了蛮力\(\left (\begin {array}{ cc}{n+k}\\ k \end {array}\right )\) - 时间搜索。

• 如果\(I=\left \langle {p_{1}(x_{1}), \ldots , p_{k}(x_{k})}\right \rangle \)，以k为固定参数，则理想成员资格测试是 W[2]-hard。当\(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{k}^{e_{k}}} \right \rangle \)。