  In this paper we study singularities in arbitrary characteristic. We propose
Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies
in terms of jet schemes of a singularity. The conjecture is equivalent to the
boundedness of the number of the blow-ups to obtain a prime divisor which
computes the Mather-Jacobian minimal log discrepancy. We also show that this
conjecture yields some basic properties of singularities; e.g. openness of
Mather-Jacobian (log) canonical singularities, stability of these singularities
under small deformations and lower semi-continuity of Mather-Jacobian minimal
log discrepancies, which are already known in characteristic 0 and open for
positive characteristic case.We show some evidences of the conjecture: for
example, for non-degenerate hypersurfaces of any dimension in arbitrary
characteristic and 2-dimensional singularities in characteristic not 2. We aslo
give a bound of the number of the blow-ups to obtain a prime divisor which
computes the Mather-Jacobian minimal log discrepancy.
