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 algebraic ...

, 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 (mathematics), homology cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map ...

complex of

f

is isomorphic to the

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

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 positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said ...

, 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