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Using the polynomial method in additive number theory, this articleestablishes a new addition theorem for the set of subsums of a set satisfying$A\cap(-A)=\emptyset$ in $\mathbb{Z}/p\mathbb{Z}$:\[|\Sigma(A)|\geqslant\min{p,1+\frac{|A|(|A|+1)}{2}}.\]  The proof is similar in nature to Alon, Nathanson and Ruzsa's proof of theErd\"os-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune\cite{DH}). A key point in the proof of this theorem is the evaluation of somebinomial determinants that have been studied in the work of Gessel and Viennot.A generalization to the set of subsums of a sequence is derived, leading to astructural result on zero-sum free sequences. As another application, it isestablished that for any prime number $p$, a maximal zero-sum free set in$\mathbb{Z}/p\mathbb{Z}$ has cardinality the greatest integer $k$ such that\[\frac{k(k+1)}{2}<p,\] proving a conjecture of Selfridge from 1976.

An addition theorem and maximal zero-sum free sets in Z/pZ