Plenty of Morse functions by perturbing with sums of squares
aa r X i v : . [ m a t h . DG ] N ov PLENTY OF MORSE FUNCTIONS BY PERTURBING WITHSUMS OF SQUARES
A. LERARIO
Abstract.
We prove that given a smooth function f : R n → R and a sub-manifold M ⊂ R n , then the set of a = ( a , . . . , a n ) ∈ R n such that ( f + q a ) | M is Morse, where q a ( x ) = a x + · · · + a n x n , is a residual subset of R n . A stan-dard transversality argument seems not to work and we need a more refinedapproach. Introduction
The following problem was posed to the author by K. Kurdyka, to which weexpress our gratitude for stimulating discussions.Given a = ( a , . . . , a n ) ∈ R n we define the function q a : R n → R by q a ( x ) = a x + · · · + a n x n where in the case b ∈ R k , k = n, we mean q b to belong to C ∞ ( R k , R ) and to bedefined in the same similar way.Suppose f ∈ C ∞ ( R n , R ) and M ⊂ R n is a submanifold. Is it true that the set A ( f, M ) = { a ∈ R n | ( f + q a ) | M is Morse } is residual in R n ?The answer to this question turns out to be affirmative, but in a subtle way: stan-dard transversality arguments based on dimension counting do not work and wehave to prove it directly.2. Failure of parametric transversality argument
We describe here what it is the usual procedure to prove that given a family offunctions f a : M → R depending smoothly on the parameter a ∈ A then the set of a such that f a is Morse is residual in A. Let G : A × M → N be a smooth map and for every a ∈ A let g a : M → N bethe function defined by x G ( a, x ) . Suppose that Z ⊂ N is a submanifold andthat F is transverse to Z. Then from the parametric transversality theorem (see [2],Theorem 2.7) it follows that { a ∈ A | g a is transverse to Z } is residual in A. In the case we want to get Morse condition consider N = T ∗ M, G ( a, x ) = d x f a and Z ⊂ T ∗ M the zero section. Then f a is Morse if and only if g a is transverse to Z. In our case, letting M = R n , we are led to define G : R n × R n → R n by( a, x ) ( ∂f /∂x ( x ) + a x , . . . , ∂f /∂x n ( x ) + a x n )and we consider Z = { } ∈ R n . Then f a = f + q a is Morse if and only if g a is transversal to { } . A condition that would ensure this (trough the parametric
SISSA, Trieste. transversality theorem) is that G is transverse to { } . Computing the differentialof G at the point ( a, x ) we have for ( v, w ) ∈ T ( a,x ) ( R n × R n )( d ( a,x ) G )( v, w ) = He( f )( x ) v + diag( a , . . . , a n ) v + diag( x , . . . , x n ) w and we see that in general this condition does not hold (for example let f ≡ , then at alle the points ( a, x ) = (0 , a , . . . , a n , , . . . ,
0) we have G ( a, x ) = 0 butrk( d ( a,x ) G ) < n ). 3. A direct approach
First we recall the following Lemma (see [1]).
Lemma 1.
Let f be a smooth function on R n and for a ∈ R n define the function f a by x f ( x ) + a x + . . . + a n x n . The set { a ∈ R n | f a is Morse } is residual in R n . Proof.
Define the function g ( x ) = ( ∂f /∂x , . . . , ∂f /∂x n ) and notice that the Hes-sian of f is precisely the Jacobian of g and that x is a nondegenerate critical pointfor f if and only if g ( x ) = 0 and the Jacobian J ( g )( x ) of g at x is nonsingular.Then g a ( x ) = g ( x ) + a and J ( g a ) = J ( g ) . We have that x is a critical point for f a if and only if g ( x ) = − a ; moreover it is a nondegenerate critical point if and onlyif we also have J ( g )( x ) is nonsingular, i.e. a is a regular value of g. The conlusionfollows by Sard’s lemma. (cid:3)
We immediately get the following corollary.
Corollary 2. If f is a smooth function on an open subset U of R n such that forevery u = ( u , . . . , u n ) ∈ U we have u i = 0 for all i = 0 , . . . , n, then A ( f, U ) = { a ∈ R n | f + q a is Morse on U } is a residual subset of R n . Proof.
The functions u , . . . , u n are coordinates on U by hypothesis; we let ˜ f bethe function f in these coordinates (it is defined on a certain open subset W of R n ). Then for every a ∈ R n we have that (using the above notation) ˜ f a is Morseon W if and only if f + q a is Morse on U and the conclusion follows applying theprevious lemma. (cid:3) To prove the general statement we need the following.
Lemma 3.
Let f be a smooth function on an (arbitrary) open subset U of R n . Then the set A ( f, U ) is residual in R n .Proof. For every I = { i , . . . , i j } ⊂ { , . . . , n } define H I = U ∩ { u i = 0 , i ∈ I } ∩ { u k = 0 , k / ∈ I } . To simplify notations let I = { , . . . , j } . Notice that if a = ( a , . . . , a n ) and a ′′ =( a j +1 , . . . , a n ) then ( q a ) | H I = ( q a ′′ ) | H I where q a ′′ : R n − j → R is defined as above.By corollary 2 the set A ′′ ( f, H I ) = { a ′′ ∈ R n − j | f | H I + q a ′′ is Morse on H I } is residual in R n − j . Let a = ( a ′ , a ′′ ) ∈ R n such that a ′′ ∈ A ′′ ( f, H I ) and suppose x ∈ H I is a critical point of f + q a ; then x is also a critical point of ( f + q a ) | H I = f | H I + q a ′′ . Since a ′′ ∈ A ′′ ( f, H I ) then x belongs to a countable set, namely the set C a ′′ of critical points of f | H I + q a ′′ (each of this critical point must be nondegenerateby the choice of a ′′ ); moreover we have thatHe( f | H I + q a ′′ )( x ) = He( f | H I )( x ) + diag( a j +1 , . . . , a n )is nondegenerate. Notice that the Hessian of f + q a at x is a block matrix:He( f + q a )( x ) = (cid:18) diag( a , . . . , a j ) + B ( x ) C ( x ) C ( x ) T He( f | H I + q a ′′ )( x ) (cid:19) . Thus for every a ′′ = ( a j +1 , . . . , a n ) ∈ A ′′ ( f, H I ) and for every x ∈ C a ′′ consider thepolynomial p a ′′ ,x ∈ R [ t , . . . , t j ] defined by p a ′′ ,x ( t , . . . , t j ) = det(He( f )( x ) + diag( t , . . . , t j , a j +1 , . . . , a n ))Then the term of maximum degree of p a ′′ ,x is t · · · t j det(He( f | H I + q a ′ )( x ))which is nonzero since det(He( f | H I + q a ′′ )( x )) = 0 ( x is a nondegenerate criticalpoint of f | H I + q a ′′ ). It follows that p a ′′ ,x is not identically zero; hence its zero locusis a proper algebraic set. Thus for each a ′′ ∈ A ′′ ( f, H I ) and each x ∈ C a ′′ the set A ′ ( a ′′ , x, I ) defined by { a ′ ∈ R j | if x is a critical point of f + q ( a ′ ,a ′′ ) on H I then it is nondegenerate } is residual in R j (it is the complement of a proper algebraic set); it follows that A ′ ( a ′′ , I ) = { a ′ ∈ R j | each critical point of f + q ( a ′ ,a ′′ ) on H I is nondegenerate } is residual in R j , since it is a countable intersection of residual sets, i.e. A ′ ( a ′′ , I ) = \ x ∈ C a ′′ A ′ ( a ′′ , x, I )Thus the set A ( f, I ) = { ( a ′ , a ′′ ) | a ′′ ∈ A ′′ ( f, H I ) , a ′ ∈ A ′ ( a ′′ , I ) } (which coincides with the set of a = ( a ′ , a ′′ ) ∈ R n such that each critical point of f + q a on H I is nondegenerate) is residual: is residual in a ′ for every a ′′ belongingto a residual set. Finally A ( f, U ) = \ I ⊂{ ,...,n } A ( f, I )is a finite intersection of residual sets, hence residual. (cid:3) Theorem 4.
Let f be a smooth function on R n and M ⊂ R n be a submanifold.Then the set A ( f, M ) is residual in R n .Proof. We basically improve the proof of Proposition 17.18 of [1].Let u , . . . , u n : R n → R be the coordinates on R n . Suppose M is of dimension m. For every point x ∈ M there exists a neighborhood W of x in M such that u i , . . . , u i m are coordinates for M on W ≃ R m , for some { i , . . . , i m } ⊆ { , . . . , n } ; since M is second countable, then it can becovered by a countable (finite if M is compact) number of such open sets. Forconvenience of notations suppose { i , . . . , i m } = { , . . . , m } . A. LERARIO
Thus u , . . . , u m are coordinates on W ≃ R m and f | W , u m +1 | W , . . . , u n | W are func-tions of u | W , . . . , u m | W . Fix a ′′ = ( a m +1 , . . . , a n ) ∈ R n − m and define g a ′′ : W → R by g a ′′ = f | W + a m +1 u m +1 | W + · · · + a n u m | W = ( f + a m +1 u m +1 + · · · + a n u n ) | W Notice that g a ′′ is not ( f + q a ) | W since we are taking only the last n − m of the a ′ i s ;we still have the freedom of choice ( a , . . . , a m ) . By lemma 3, since u | W , . . . , u m | W are coordinates on W , for every a ′′ ∈ R n − m theset { a ′ = ( a , . . . , a m ) ∈ R m s.t. g a ′′ + a u | W + · · · + a m u m | W is Morse on W } is residual in R m . Notice that g a ′′ + a u | W + · · · + a m u m | W = ( f + q ( a ′ ,a ′′ ) ) | W ;hence for every a ′′ the set of a ′ such that ( f + q ( a ′ ,a ′′ ) ) | W is Morse on W is residual.Thus the set of a ∈ R n such that ( f + q a ) | W is Morse on W is residual (it is residualin a ′ for each fixed a ′′ hence it is globally residual). It follows that A ( f, M ) is acountable intersection of residual set, hence residual. (cid:3) References [1] R. Bott and L. Tu:
Differential Forms in Algebraic Topology , Springer-Verlag, 1982.[2] M.W. Hirsch:
Differential Topology , Springer-Verlag, 1997.[3] A. Lerario:
Systems of two quadratic inequalities , arXiv:1106.4678v1[4] J. Milnor,