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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 ...

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