# Splitting lemma (functions)

*See also splitting lemma in homological algebra.*

In mathematics, especially in singularity theory the **splitting lemma** is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point.

## Formal statement

Let be a smooth function germ, with a critical point at 0 (so ). Let *V* be a subspace of such that the restriction *f|V* is non-degenerate, and write *B* for the Hessian matrix of this restriction. Let *W* be any complementary subspace to *V*. Then there is a change of coordinates of the form with , and a smooth function *h* on *W* such that

This result is often referred to as the parametrized Morse lemma, which can be seen by viewing *y* as the parameter.
It is the *gradient version* of the implicit function theorem.

## Extensions

There are extensions to infinite dimensions, to complex analytic functions, to functions invariant under the action of a compact group, . . .

## References

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