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If you have $t-1$ shares of an $(t,n)$ system, you have a chance at learning of the roots of the system... as long as some of your shares have a 0 for the $y$ coordinate. You can't learn any other roots. Demonstration that the possession of a share with a $y$ coordinate of 0 gives you knowledge of a root (and forgive me if this is too obvious): A share ...


Just to be specific we are talking about information theoretically secure secret sharing, e.g. Shamir sharing. Lets say a secret $secret$ is shared in $n$ shares $S = \{s_1, \ldots, s_n\}$. Assume further party $P_i$ is given share $s_i$. If I understand your question correctly, you are asking if we give $P_i$ some share $r \ne s_i$ is there anyway for $P_i$ ...

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