Counting polynomial subset sums

Abstract

Let D be a subset of a finite commutative ring R with identity. Let f (x) ∈ R[x] be a polynomial of positive degree d. For integer 0 ≤ k ≤ |D|, we study the number N f (D, k, b) of k-subsets S ⊆ D such that x∈Swhere e(G) is the exponent of G and δ(n) = i|n,µ(i)=−1 1 i . In particular, we give a new short proof for the explicit counting formula for the case D = G.

Moment subset sums over finite fields

“…Following established procedures, we use the Li-Wan sieve [17] to analyze large values of k. This method has been used several times [26,14,17,18,19,24] and is now standard. So, we will only give an outline and indicate the differences.…”

Applications Of The Subset Sum Problem Over Finite Abelian Groups

“…where S r is the set of permutations of {1, 2, ..., r}, l(σ) is the number of disjoint cycles in σ including length 1, and F σ is the number of r-tuples (x 1 , x 2 , ..., x r ) ∈ D r where r j=1 f (x j ) = b such that x i = x j whenever i and j belong to the same cycle in σ. Li and Wan [10] used their sieve formula to find the number of r-subsets of a finite abelian group G whose elements sum to b ∈ G. These authors [9] found several bounds for S G D r,f (x) , b when G is the additive group of a commutative ring R with identity, D ⊆ G, and f ∈ R [x].…”