  Using the polynomial method in additive number theory, this article
establishes 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 the
Erd\"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 some
binomial 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 a
structural result on zero-sum free sequences. As another application, it is
established 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.
