Sums of subsets with polynomial restrictions Original Research Article

Let F be a field of characteristic p and let P(x)set membership, variantF[x] be a polynomial of degree m>0. Let A1,…,An be finite subsets of F with An=k>m(n−1) and Ai+1−Aiset membership, variant{0,1} for i=1,…,n−1. If p=0 or image, then for the restricted sumsetimageS={a1+cdots, three dots, centered+an: a1set membership, variantA1,…,anset membership, variantAn, and P(ai)≠P(aj) if i≠j}we have image. This extends the Erdös–Heilbronn conjecture posed in 1964 and confirmed in 1994.