In this paper we develop an efficient procedure for computing a (scaled) Hadamard product for commutative polynomials. This serves as a tool to obtain faster algorithms for several problems. Our main algorithmic results include the following:

Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f ∈ F[X], where X = {x1, x2, . . . , xn} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [13,22,14].(k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f . k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f . Our algorithms are based on the fact that the Hadamard product f • S n,k , is the degree-k multilinear part of f , where S n,k is the k th elementary symmetric polynomial. For (k,n)-MLC problem, we give a deterministic algorithm of run time O * (n k/2+c log k ) (where c is a constant), answering an open question of Koutis and Williams [14, ICALP'09]. As corollaries, we show O * ( n ↓k/2

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We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the rectangular symbolic matrix in both commutative and noncommutative settings. The main results are:We show an explicit O * ( n ↓k/2 )-size ABP construction for noncommutative permanent polynomial of k×n symbolic matrix. We obtain this via an explicit ABP construction of size O * ( n ↓k/2 ) for S * n,k , noncommutative symmetrized version of the elementary symmetric polynomial S n,k . We obtain an explicit O * (2 k )-size ABP construction for the commutative rectangular determinant polynomial of the k × n symbolic matrix. In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard.

Let F[X] be the polynomial ring over the variables X = {x 1 , x 2 , . . . , x n }. An ideal I = p 1 (x 1 ), . . . , p n (x n ) generated by univariate polynomials {p i (x i )} n i=1 is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results.• Let f (X) ∈ F[ℓ 1 , . . . , ℓ r ] be a (low rank) polynomial given by an arithmetic circuit where ℓ i : 1 ≤ i ≤ r are linear forms, and I = p 1 (x 1 ), . . . , p n (x n ) be a univariate ideal. Given α ∈ F n , the (unique) remainder f (X) (mod I) can be evaluated at α in deterministic time d O(r) • poly(n), where d = max{deg(f ), deg(p 1 ) . . . , deg(p n )}. This yields an 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 × 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 [6] via a different technique.• Let f (X) ∈ F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I = p 1 (x 1 ), . . . , p n (x n ) . We show that in the special case when I = x e1 1 , . . . , x en n , we obtain a randomized O * (4.08 k ) algorithm that uses poly(n, k) space.• Given f (X) ∈ F[X] by an arithmetic circuit and I = p 1 (x 1 ), . . . , p k (x k ) , membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I = x e1 1 , . . . , x e k k .