Mutations of potentials and their upper bounds

We introduce Berenstein's notion of upper bounds to theory of mutations of potentials and prove an excessive Laurent phenomenon (which now says - upper bounds are preserved by mutations) using this new technique. The statement is quite general and can be applied in different contexts where Laurent polynomials and their mutations appear (e.g. Auroux's wall-crossing in symplectic geometry and mirror symmetry). This is a joint work with John Alexander Cruz Morales (preprint IPMU 12-0110).