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

Let 𝔽[X] be the polynomial ring in the variables X = x 1 , x 2 ,…, x n over a field 𝔽 . An ideal I = 〈 p 1 ( x 1 ),…, 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∈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 )mod I at any point a⃗∈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 𝔽 . Let f be over rationals with (f)=k treated as fixed parameter. When the ideal I=⟨x_1^e_1, … , x_n^e_n⟩ , we can test ideal membership in randomized O ∗ ((2 e ) 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 ([ n+k; k ] ) -time search. If I=⟨p_1(x_1), … , p_k(x_k)⟩ , 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=⟨x_1^e_1, … , x_k^e_k⟩ .