Critical point

For an analytic function $ f ( z) $, a critical point of order $ m $ is a point $ a $ of the complex plane at which $ f ( z) $ is regular but its derivative $ f ^ { \prime } ( z) $ has a zero of order $ m $, where $ m $ is a natural number. In other words, a critical point is defined by the conditions

$$ \lim\limits _ {z \rightarrow a } \frac{f ( z) - f ( a) }{( z - a) ^ {m} } = 0,\ \ \lim\limits _ {z \rightarrow a } \frac{f ( z) - f ( a) }{( z - a) ^ {m+ 1} } \neq 0. $$

A critical point at infinity, $ a = \infty $, of order $ m $, for a function $ f ( z) $ which is regular at infinity, is defined by the conditions

$$ \lim\limits _ {z \rightarrow \infty } [ f ( z) - f ( \infty )] z ^ {m} = 0,\ \ \lim\limits _ {z \rightarrow \infty } [ f ( z) - f ( \infty )] z ^ {m + 1 } \neq 0. $$

Under the analytic mapping $ w = f ( z) $, the angle between two curves emanating from a critical point of order $ m $ is increased by a factor $ m + 1 $. If $ f ( z) $ is regarded as the complex potential of some planar flow of an incompressible liquid, a critical point is characterized by the property that through it pass not one but $ m + 1 $ stream lines, and the velocity of the flow at a critical point vanishes. In terms of the inverse function $ z = \psi ( w) $ (i.e. the function for which $ f [ \psi ( w)] \equiv w $), a critical point is an algebraic branch point of order $ m + 1 $.

A point $ a $ of a complex $ ( n - m) $-dimensional irreducible analytic set

$$ M = \ \{ {z \in V } : { f _ {1} ( z) = \dots = f _ {m} ( z) = 0 } \} , $$

the latter being defined in a neighbourhood $ V $ of $ a $ in the complex space $ \mathbf C ^ {n} $ by the conditions $ f _ {1} ( z) = \dots = f _ {m} ( z) = 0 $, where $ f _ {1}, \dots, f _ {m} $ are holomorphic functions on $ V $ in $ n $ complex variables, $ z = ( z _ {1}, \dots, z _ {n} ) $, is called a critical point if the rank of the Jacobian matrix $ \| \partial f _ {j} / \partial z _ {k} \| $, $ j = 1, \dots, m $, $ k = 1, \dots, n $, is less than $ m $. The other points of $ M $ are called regular. There are relatively few critical points on $ M $: They form an analytic set of complex dimension at most $ n - m - 1 $. In particular, when $ m = 1 $, i.e. if $ M = \{ f _ {1} ( z) = 0 \} $, and the dimension of $ M $ is $ n - 1 $, the dimension of the set of critical points is at most $ n - 2 $.

References

[1]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) MR0173749 Zbl 0135.12101
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001

Comments

A point as described under 2) is also called a singular point of $ M $, cf. [a1].

References

[a1]	H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) pp. 95 (Translated from German) MR0414912 Zbl 0381.32001

A critical point of a smooth (i.e. continuously differentiable) mapping $ f $ of a $ k $-dimensional differentiable manifold $ M $ into an $ l $-dimensional differentiable manifold $ N $ is a point $ x _ {0} \in M $ such that the rank $ \mathop{\rm Rk} _ {x _ {0} } f $ of $ f $ at this point (i.e. the dimension of the image $ df ( T _ {x _ {0} } M) $ of the tangent space to $ M $ under the differential mapping $ df: T _ {x _ {0} } M \rightarrow T _ {f ( x _ {0} ) } N $) is less than $ l $. The set of all critical points is called the critical set, the image $ f( x _ {0} ) $ of a critical point $ x _ {0} $ is called a critical value, and a point $ y \in N $ which is not the image of any critical point is called a regular point or a regular value (though it need not belong to the image $ f( M) $); non-critical points of $ M $ are also called regular.

According to Sard's theorem, if $ f $ is smooth of class $ C ^ {m} $, $ m > \min ( k - l, 0) $, then the image of the critical set is of the first category in $ N $ (i.e. it is the union of at most countably many nowhere-dense sets) and has $ l $-dimensional measure zero (see [1], [2]). The lower bound for $ m $ cannot be weakened (see [3]). The case most frequently needed is $ m = \infty $ (in which case the proof is simplified, see [4]). This theorem is widely used for reductions to general position via "small movements"; for example, it may readily be used to prove that, given two smooth submanifolds in $ \mathbf R ^ {n} $, there exists an arbitrarily small translation of one of them such that their intersection will also be a submanifold (see [2], [4], and also Transversality of mappings).

According to the above definition, when $ k < l $ every point $ x _ {0} \in M $ must be considered as critical. Then, however, there are considerable differences between the properties of the points $ x _ {0} $ for which $ \mathop{\rm Rk} _ {x _ {0} } f = k $ and the points for which $ \mathop{\rm Rk} _ {x _ {0} } f < k $. In the former case there is a neighbourhood of $ x _ {0} $ in which the mapping $ f $ looks roughly like the standard imbedding of $ \mathbf R ^ {k} $ into $ \mathbf R ^ {l} $; more precisely, there exist local coordinates $ x _ {1}, \dots, x _ {k} $ near $ x _ {0} $ (on $ M $) and $ y _ {1}, \dots, y _ {l} $ near $ f ( x _ {0} ) $ (on $ N $), in terms of which $ f $ is given by

$$ y _ {i} = x _ {i} ,\ \ i \leq k; \ \ y _ {k + 1 } = \dots = y _ {l} = 0. $$

In the second case the image of a neighbourhood of $ x _ {0} $ need not be a manifold, displaying instead various singularities — cusps, self-intersections, etc. For this reason, the definition of a critical point is often modified to include only points $ x _ {0} $ such that $ \mathop{\rm Rk} _ {x _ {0} } f < \min ( k, l) $; corresponding modifications are then necessary in the definitions of the other terms listed above [5].

The behaviour of mappings in a neighbourhood of a critical point is investigated in the theory of singularities of differentiable mappings (see [5], [6]). In that context one studies not arbitrary critical points (concerning which little can be said), but critical points satisfying conditions which ensure that they are "not too strongly degenerate" and "typical". Thus, one considers critical points of sufficiently smooth mappings, or families of mappings (which depend smoothly on finitely many parameters), which are "unremovable" in the sense that, under small perturbations ("small" being understood in the sense of $ C ^ {m} $ for suitable $ m $) of the original mapping, or of the original family, the perturbed mapping (family) has a critical point of the same type in some neighbourhood of the original critical point. For a mapping $ M \rightarrow \mathbf R $ (i.e. an ordinary scalar function; in this case critical points are often called stationary points), critical points which are typical in the indicated sense are the non-degenerate critical points at which the Hessian is a non-degenerate quadratic form. Concerning typical critical points for a family of functions see [6], [7].

References

[1]	A. Sard, "The measure of critical values of differentiable maps" Bull. Amer. Math Soc. , 48 (1942) pp. 883–890 MR7523 Zbl 0063.06720
[2]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
[3]	H. Whitney, "A function not constant on a connected set of critical points" Duke Math. J. , 1 : 4 (1935) pp. 514–517 MR1545896 Zbl 0013.05801 Zbl 61.1117.01 Zbl 61.0262.07
[4]	J.W. Milnor, "Topology from the differential viewpoint" , Univ. Virginia Press (1965)
[5]	M. Golubitsky, "Stable mappings and their singularities" , Springer (1974) MR0467801 MR0341518 Zbl 0434.58001 Zbl 0429.58004 Zbl 0294.58004
[6]	P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) MR0494220 Zbl 0302.58006
[7]	V.I. Arnol'd, "Normal forms of functions near degenerate critical points, the Weyl groups $A_k$, $D_k$, $E_k$ and Lagrangian singularities" Funktsional. Anal. i Prilozh. , 6 : 4 (1972) pp. 3–25 (In Russian)

D.V. Anosov