Singularity confinement and projective resolution of triangulated category

Derivation of higher dimensional periodic recurrence equations by nested structure of complex numbers

“…Furthermore, [12] suggested that a derivation of sequence of IVPPs from SC is interpreted as projective resolution of singularity of a map in a triangulated category. We reinterpret the fact by our results that a map can be expanded by a sequence of PREs like a Fourier expansion, i.e., a map X = F (x) which has IVPP can be written as…”

On Incompleteness of Some Integrable Rational Maps

“…3. We have investigated many integrable rational maps satisfying p ≥ d/2, and found IVPPs of all peiods for each map iteratively by a projective resolution of the singularity [9,10,11,12].…”

“…2b. It is not difficult to find the curves of selfintersection of the surface (12). They consist of two independent pieces…”

“…We can derive IVPPs of all periods iteratively using the method developed in [9,10,12]. For example we find γ's of (6) as…”

“…This means that the singular point is a 'fixed point' of the map. We have shown in [11,12] that this phenomenon can be associated with the projective resolution of a triangulated category.…”