# Splitting lemma (functions)

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 $f:({\mathbb {R} }^{n},0)\to ({\mathbb {R} },0)$ be a smooth function germ, with a critical point at 0 (so $(\partial f/\partial x_{i})(0)=0,\;(i=1,\dots ,n)$ ). Let V be a subspace of $\mathbb {R} ^{n}$ 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 $\Phi (x,y)$ of the form $\Phi (x,y)=(\phi (x,y),y)$ with $x\in V,\;y\in W$ , and a smooth function h on W such that

$f\circ \Phi (x,y)=\textstyle {\frac {1}{2}}x^{T}Bx+h(y).$ 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, . . .