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

Authors:	Peter Borwein Michael J Mossinghoff

Institution:	Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 ; Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095

Abstract:	We study the problem of determining the minimal degree of a polynomial that has all coefficients in $\{0,1\}$ and a zero of multiplicity at . We show that a greedy solution is optimal precisely when $m\leq5$ , mirroring a result of Boyd on polynomials with $\pm1$ coefficients. We then examine polynomials of the form $\prod_{e\in E} (x^e+1)$ , where is a set of positive odd integers with distinct subset sums, and we develop algorithms to determine the minimal degree of such a polynomial. We determine that satisfies inequalities of the form $2^m + c_1 m \leq d(m) \leq \frac{103}{96}\cdot2^m + c_2$ . Last, we consider the related problem of finding a set of positive integers with distinct subset sums and minimal largest element and show that the Conway-Guy sequence yields the optimal solution for $m\leq9$ , extending some computations of Lunnon.

Keywords:	Newman polynomial pure product sum-distinct sets Conway-Guy sequence

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