complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...

, a branch of mathematics, the Thom–Sebastiani Theorem states: given the

germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embry ...

f : (\mathbb^, 0) \to (\mathbb, 0)

defined as

f(z_1, z_2) = f_1(z_1) + f_2(z_2)

where

f_i

are germs of

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

s with isolated singularities, the

vanishing cycle In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map from a connected c ...

complex of

f

is isomorphic to the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

of those of

f_1, f_2

. Moreover, the isomorphism respects the monodromy operators in the sense:

T_ \otimes T_ = T_f

. The theorem was introduced by Thom and Sebastiani in 1971. Observing that the analog fails in

positive characteristic In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...

, Deligne suggested that, in positive characteristic, a tensor product should be replaced by a (certain) local convolution product.

References

* Theorems in complex analysis {{DEFAULTSORT:Thom-Sebastiani Theorem