In commutative algebra, the deviations of a local ring R are certain invariants õ<sub>i</sub>(R) that measure how far the ring is from being regular.
The deviations õ<sub>n</sub> of a local ring R with residue field k are non-negative integers defined in terms of its Poincaré series P(t) by
The zeroth deviation õ<sub>0</sub> is the embedding dimension of R (the dimension of its tangent space). The first deviation õ<sub>1</sub> vanishes exactly when the ring R is a regular local ring, in which case all the higher deviations also vanish. The second deviation õ<sub>2</sub> vanishes exactly when the ring R is a complete intersection ring, in which case all the higher deviations vanish.