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For m by n partial totally positive matrices with exactly one unspeci ed entry the set of positions for that entry that guarantee completability to a totally positive matrix are characterized They are the positions i j i j and the positions i j i j m n In each case the set of completing entries is an open and in nite in case i j or i m j n interval In the process some new structural results about totally positive matrices are developed In addition the pairs of positions that guarantee completability in partial totally positive matrices with two unspeci ed entries are characterized in low dimensions

The general totally positive matrix completion problem with few unspecified entries