Nash problem for a toric pair and the minimal log-discrepancy

This paper formulates the Nash problem for a pair consisting of a toric variety and an invariant ideal and gives an affirmative answer to the problem. We also prove that the minimal log-discrepacy is computed by a divisor corresponding to a Nash component, if the minimal log-discrepancy is finite. On the other hand there exists a Nash component such that the corresponding divisor has negative log-discrepancy, if the minimal log-discrepancy is $-/infty$.