Generation problems

Here, the class of bivariate polynomials with positive coefficients turns out to be the most interesting class of operations. We show that many of the corresponding generation problems belong to NP. However, we do not know this for all of them, e.g., for x2+2y this is an open question. We prove NP-completeness for polynomials xaybc where a,b,c⩾1. Also, we show NP-hardness for polynomials like x2+2y. As a by-product we obtain NP-completeness of the extended sum-of-subset problem SOSc={(w1,…,wn,z):∃I⊆{1,…,n}(∑i∈Iwic=z)} for any c⩾1.