Result: Newman polynomials with prescribed vanishing and integer sets with distinct subset sums

We study the problem of determining the minimal degreed(m)d(m)of a polynomial that has all coefficients in{0,1}\{0,1\}and a zero of multiplicitymmat−1-1. We show that a greedy solution is optimal precisely whenm≤5m\leq 5, mirroring a result of Boyd on polynomials with±1\pm 1coefficients. We then examine polynomials of the form∏e∈E(xe+1)\prod _{e\in E} (x^e+1), whereEEis a set ofmmpositive odd integers with distinct subset sums, and we develop algorithms to determine the minimal degree of such a polynomial. We determine thatd(m)d(m)satisfies inequalities of the form2m+c1m≤d(m)≤10396⋅2m+c22^m + c_1 m \leq d(m) \leq \frac {103}{96}\cdot 2^m + c_2. Last, we consider the related problem of finding a set ofmmpositive integers with distinct subset sums and minimal largest element and show that the Conway-Guy sequence yields the optimal solution form≤9m\leq 9, extending some computations of Lunnon.