Relating Catlin and D'Angelo $q$-types
Vasile Brinzanescu Andreea C. Nicoara
math.CV math.AG 32F18, 32T25 (Primary), 32V35, 13H15 (Secondary)

Relating Catlin and D'Angelo q-types]Relating Catlin and D'Angelo q-types


Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 3, 21 Calea Grivitei Street, 010702 Bucharest, Romania

Vasile.Brinzanescu@imar.ro




School of Mathematics, Trinity College Dublin, Dublin 2, Ireland

anicoara@maths.tcd.ie

[2020]Primary 32F18; 32T25; Secondary 32V35; 13H15.








We clarify the relationship between the two most standard measurements of the order of contact of q-dimensional complex varieties with a real hypersurface, the Catlin and D'Angelo q-types, by showing that the former equals the generic value of the normalized order of contact measured along curves whose infimum is by definition the D'Angelo q-type.  



[
    Andreea C. Nicoara
    May 1, 2019
======================








§ INTRODUCTION



The purpose of this note is finishing the work initiated in <cit.> as far as elucidating the relationship between Catlin and D'Angelo q-types. D'Angelo's finite q-type defined in 1982 in <cit.> is by far the most standard finite type notion in several complex variables. The Kohn Conjecture, one of the most famous open problems in complex analysis, if proven would say that finite D'Angelo q-type at a certain point on the boundary of a smooth pseudoconvex domain in ^n ensures the termination of the Kohn algorithm defined in <cit.> and thus the subellipticity of the -Neumann problem for (p,q) forms in the neighborhood of that point. As of now, the only result on the subellipticity of the -Neumann problem for (p,q) forms on smooth pseudoconvex domains in ^n is due to Catlin in  <cit.> and provides a lower bound on the subelliptic gain for the -Neumann problem in terms of his own notion of q-type. As a result, relating the D'Angelo and Catlin q-types effectively is paramount so that if and when the Kohn Conjecture is proven with a lower bound for subellipticity in place, Catlin's bound and the bound obtained via the Kohn algorithm can be compared.




As the results in <cit.> are proven with respect to the generic value, we introduce the following definition; see also <cit.>:




Let 2 ≤ q ≤ n. If  is an ideal in _x_0,

    Δ̃_q(, x_0)=gen.val_{w_1, …, w_q-1}   Δ_1( (, w_1, …, w_q-1), x_0 ),
 where the generic value is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0, (, w_1, …, w_q-1) is the ideal in _x_0 generated by , w_1, …, w_q-1, and Δ_1 is the D'Angelo 1-type. Likewise, if M is a real hypersurface in ^n and x_0 ∈ M, 
    Δ̃_q (M, x_0) =gen.val_{w_1, …, w_q-1}   Δ_1( ((M), w_1, …, w_q-1), x_0 ),
 where ((M), w_1, …, w_q-1) is the ideal in C^∞_x_0 generated by all smooth functions (M) vanishing on M along with w_1, …, w_q-1. 




Let D_q denote the Catlin q-type, and let Δ_q be the D'Angelo q-type. The two main theorems in <cit.> can now be stated with respect to Δ̃_q as follows:

[Theorem 1.1 <cit.>,<cit.>]

Let  be an ideal of germs of holomorphic functions at x_0, then for 1 ≤ q ≤ n 
    D_q(, x_0) ≤Δ̃_q(, x_0) ≤(D_q(, x_0))^n-q+1.



[Theorem 1.2 <cit.>,<cit.>]

Let Ω in ^n be a domain with 
boundary. Let x_0 ∈ b Ω be a point on the boundary of
the domain, and let 1 ≤ q <n.


  * D_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0);

  * If Δ̃_q(b Ω, x_0)<∞ and the domain is q-positive at x_0 (the q version of D'Angelo's property P), then 
    Δ̃_q(b Ω, x_0) ≤ 2 ( D_q(b Ω, x_0)/2)^n-q.


 In particular, if b Ω is pseudoconvex at x_0 and Δ̃_q(b Ω, x_0)<∞, then 
    D_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0) ≤ 2 ( D_q(b Ω, x_0)/2)^n-q.




It is an easy consequence of D'Angelo's work in <cit.> that Δ_q and Δ̃_q are simultaneously finite for ideals of germs of holomorphic functions:




Let  be any ideal in _x_0. For any 2 ≤ q ≤ n, 
    Δ_q(, x_0) ≤Δ̃_q(, x_0) ≤( Δ_q(, x_0) )^n-q+1.



The counterpart of this result for points on the boundary of a domain is slightly more difficult and requires the assumption of q-positivity:



Let Ω in ^n be a domain with 
boundary. Let x_0 ∈ b Ω be a point on the boundary of
the domain, and let 2 ≤ q <n. If the domain is q-positive at x_0, then

    Δ_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0) ≤ 2 ( Δ_q(b Ω, x_0) )^n-q.
 If the domain is pseudoconvex at x_0, then it is q-positive, so the same inequality holds.



We shall prove Propositions <ref> and  <ref> in Section <ref>. We note here at for q=1 the notions of Δ_q, Δ̃_q, and D_q coincide, a fact obvious from their definitions.

Martino Fassina produced an example in <cit.> when the Catlin and D'Angelo q-types are not equal to each other, answering a question that has been open since 1987 when <cit.> was published. In that example, 
    Δ_2(,0)= 3 < 4 = Δ̃_2(,0) = D_2(,0).
 Furthermore, in the same paper <cit.>, Fassina proved that given any positive integer, an ideal of holomorphic germs can be constructed so that the difference between the Catlin q-type and the D'Angelo q-type of that ideal is larger than the given integer. Fassina's work thus highlights the importance of finding an effective relationship between Δ_q and D_q.

The main result of this paper is the following characterization of the Catlin q-type D_q that clarifies completely how it relates to the D'Angelo q-type:






  * Let  be an ideal of germs of holomorphic functions at x_0, and let 2≤ q ≤ n, then 
    D_q(, x_0) = Δ̃_q(, x_0).


  * Let Ω in ^n be a domain with  boundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, and let 2 ≤ q <n. Then 
    D_q(b Ω, x_0) = Δ̃_q(b Ω, x_0).








  * Let  be an ideal of germs of holomorphic functions at x_0, and let 2≤ q ≤ n, then 
    Δ_q(, x_0) ≤ D_q(, x_0) ≤( Δ_q(, x_0) )^n-q+1.


  * Let Ω in ^n be a domain with  boundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, let 2 ≤ q <n, and assume that the domain is q-positive at x_0. Then 
    Δ_q(b Ω, x_0) ≤ D_q(b Ω, x_0) ≤ 2 ( Δ_q(b Ω, x_0) )^n-q.
 In particular, if the domain is pseudoconvex at x_0, then the same inequality holds.



In <cit.> Catlin defined his q-type D_q by starting with the germ of a q-dimensional variety V^q and constructing an open set in the Grassmannian G^n-q+1 of all (n-q+1)-dimensional complex linear subspaces through x_0 in ^n that is specific to a given V^q. His construction is so delicate because the aim is to obtain the same number of curves in the intersection of V^q with each (n-q+1)-dimensional complex linear subspace in this open set and the same maximal normalized order of contact measured along the intersection curves. As a result, proving the equality of D_q with Δ̃_q is trickier than it seems. Given any curve and any subspace W, we have to show the existence of a cylinder variety with the subspace W as the directrix subspace along the curve at x_0. This construction allows us to construct a q-dimensional variety V^q starting with any curve whose open set in the Grassmanian G^n-q+1 in Catlin's construction is well behaved.

The paper is organized as follows: We devote Section <ref> to recalling the definitions of the Catlin and D'Angelo q-types and relevant results. We then prove Propositions <ref> and <ref> as well as the Main Theorem, Theorem <ref>, and Corollary <ref> in Section <ref>.

Acknowledgements
The work of Vasile Brinzanescu was partially supported by a grant of the Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0030, within PNCDI III, and CNCS-UEFISCDI project PN-III-P4-ID-PCE-2020-0029, Syzygies, invariants and classification problems in algebraic geometry and topology.




§ CATLIN AND D'ANGELO Q-TYPES



For the convenience of the reader, we first recall the definitions of the D'Angelo and Catlin q-types and the properties needed for proving the results in Section <ref>.



Let C^∞_x_0 be the ring of smooth germs at x_0 ∈^n, and let  be an ideal in C^∞_x_0 or _x_0. 
    Δ_1(,x_0)   = sup_φ∈(n,x_0)  inf_g ∈  ord_0  φ^* g/ord_0  φ,
 where (n,x_0) is the set of all germs of holomorphic curves 
    φ: (U,0) → (^n, x_0)
 such that φ(0)=x_0 for U is some neighborhood of the origin in ^1, ord_0 is the vanishing order at the origin, and  ord_0  φ = min_1 ≤ j ≤ m ord_0  φ_j.



Let 2 ≤ q ≤ n. If  is an ideal in _x_0, the D'Angelo q-type is given by

    Δ_q(, x_0)=inf_{w_1, …, w_q-1}   Δ_1( (, w_1, …, w_q-1), x_0 ),
 where the infimum is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0 and (, w_1, …, w_q-1) is the ideal in _x_0 generated by , w_1, …, w_q-1. Likewise, if M is a real hypersurface in ^n, x_0 ∈ M, and 2 ≤ q < n, the D'Angelo q-type of the hypersurface M is given by 
    Δ_q (M, x_0) =inf_{w_1, …, w_q-1}   Δ_1( ((M), w_1, …, w_q-1), x_0 ),
 where ((M), w_1, …, w_q-1) is the ideal in C^∞_x_0 generated by all smooth functions (M) vanishing on M along with w_1, …, w_q-1. 



If  is an ideal in _x_0, 
    D(, x_0) = _ (_x_0 / ).



[Proposition 2.8 <cit.>]

Let  be a proper ideal in _x_0, and let x_0 ∈^n.

    inf_{w_1, …, w_q-1} D( (, w_1, …, w_q-1), x_0 )=gen.val_{w_1, …, w_q-1} D( (, w_1, …, w_q-1), x_0 ),

where {w_1, …, w_q-1} is a non-degenerate set of linear forms in _x_0, (, w_1, …, w_q-1) is the ideal in _x_0 generated by , w_1, …, w_q-1, and the infimum and the generic value are both taken over all such non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0. In other words, the infimum is achieved and equals the generic value.


[D'Angelo, Theorem 2.7 <cit.>]

If  is an ideal in _x_0 containing q linearly independent linear forms w_1, …, w_q, then 

    Δ_1(, x_0) ≤ D(, x_0) ≤( Δ_1(,x_0) )^n-q.



If Δ_q (M, x_0) = t < ∞, let k = ⌈ t ⌉ be the ceiling of t. By Proposition 14 from p.88 of <cit.>, 

    Δ_q (M, x_0) = Δ_q (M_k, x_0),
 
where M_k is real hypersurface defined by r_k, the polynomial with the same k-jet at x_0 as the defining function r of M. By D'Angelo's polarization technique from <cit.>, 
    r_k = Re{h} + ||f||^2-||g||^2,
 where ||f||^2= ∑_j=1^N |f_j|^2, ||g||^2= ∑_j=1^N |g_j|^2, and the functions h, f_1, …, f_N, g_1, …, g_N are all holomorphic polynomials in n variables. Let (N) be the group of N × N unitary matrices. ∀  U ∈ (N) consider the ideal of holomorphic polynomials (U, x_0)=(h, f-Ug) generated by h and the N components of f-Ug, where f=(f_1, …, f_N) and g= (g_1, …, g_N). D'Angelo proved the following result:

[Theorem 14 <cit.>]


    Δ_q(M, x_0) ≤ 2   sup_U ∈ (N)  inf_{w_1, …, w_q-1}   D( ((U,x_0), w_1, …, w_q-1), x_0 ) ≤ 2 (Δ_q(M, x_0) )^n-q



Let us also note that (N) is compact, so sup_U ∈ (N)  D( ((U,x_0), w_1, …, w_q-1) is achieved due to the fact that the multiplicity of an ideal is upper semi-continuous:

[Part of Proposition II.5.3 <cit.>]

Let (λ) be an ideal in _x_0 that depends continuously on λ. Then D((λ), x_0) is an upper semi-continuous function of λ. 



We recall from <cit.> the q version of D'Angelo's property P, q-positivity, the hypothesis that appears in Theorem <ref>:

[Definition 2.14 <cit.>]

Let M be a real hypersurface of ^n, and let x_0 ∈ M be such that Δ_q(M, x_0) <k. Let j_k,x_0 r = r_k = Re{h} + ||f||^2-||g||^2 be a holomorphic decomposition at x_0 of the k-jet of the defining function r of M. We say that M is q-positive at x_0 if for every holomorphic curve φ∈(n,x_0) for which φ^* h vanishes and such that the image of φ locally lies in the zero locus of a non-degenerate set of linear forms {w_1, …, w_q-1} at x_0, the following two conditions are satisfied:


  (i) ord_0  φ^* r is even, i.e. ord_0  φ^* r = 2a, for some a ∈;

  (ii) (d/dt)^a (d/d t̅)^a φ^* r (0) ≠ 0.



When q=1 this definition is exactly D'Angelo's property P with respect to which he proved the following result:

[Theorem 5.3 <cit.>]

If M satisfies property P at x_0, then 
    Δ_1(M, x_0) = 2  sup_U ∈ (N)  Δ_1((U,x_0), x_0).



Under the assumption of q-positivity, it is obvious that D'Angelo's result immediately implies the following:



If M is q-positive at x_0, then 
    Δ̃_q (M, x_0) = 2  gen.val_{w_1, …, w_q-1} sup_U ∈ (N)  Δ_1 (((U,x_0), w_1, …, w_q-1)),x_0).



Pseudoconvexity and q-positivity relate to each other as follows:

[Proposition 2.18  <cit.>]

If M is pseudoconvex near x_0 and Δ_q (M, x_0)< + ∞, then M and M_k, the hypersurface corresponding to the truncation of order k of the defining function at x_0, are q-positive at x_0 for all sufficiently large k.




Let V^q be the germ of a q-dimensional complex variety passing through x_0. Let G^n-q+1 be the set of all (n-q+1)-dimensional complex linear subspaces through x_0 in ^n. Consider the intersection V^q ∩ S for S ∈ G^n-q+1. For a generic, namely open and dense, subset W̃ of G^n-q+1, V^q ∩ S consists of finitely many irreducible one-dimensional components V^q_S,k for k=1, …, P. We parametrize each such germ of a curve by some open set U_k ∋ 0 in . Thus, γ_S^k : U_k → V^q_S,k, where γ_S^k (0)=x_0. For every holomorphic germ f ∈_x_0, let 
    τ (f, V^q ∩ S) = max_k=1, …, Pord_0  (γ^k_S)^* f/ord_0  γ^k_S.
 Likewise, for r the defining function of a real hypersurface M in ^n passing through x_0, let 
    τ (V^q ∩ S, x_0) = max_k=1, …, Pord_0  (γ^k_S)^* r/ord_0  γ^k_S.
 In Proposition 3.1 of <cit.>, Catlin showed τ (f, V^q ∩ S) assumes the same value for all S in a generic subset W̃ of linear subspaces that depends on V^q, so he defined 
    τ(f, V^q) = gen.val_S ∈W̃{τ (f, V^q ∩ S) }
 and 
    τ(, V^q) = min_f ∈τ(f, V^q).
 Proposition 3.1 of <cit.> likewise implies that τ (V^q ∩ S, x_0) assumes the same value for all S in the same generic subset W̃ of linear subspaces depending on V^q, so Catlin defined 
    τ(V^q, x_0) = gen.val_S ∈W̃{τ (V^q ∩ S, x_0) }.
 We will need to explain exactly how the generic subset W̃ depends on the germ of the q-dimensional complex variety V^q when we prove Theorem <ref> in Section <ref>, so we defer that discussion.




Let  be an ideal of holomorphic germs at x_0, then the Catlin q-type of the ideal  is given by 
    D_q (, x_0)=sup_V^q{τ(, V^q) }.
 
Let M be a real hypersurface in ^n, and let x_0 ∈ M. The Catlin q-type of M at x_0 is given by 
    D_q (M, x_0)=sup_V^q{τ (V^q, x_0) }.
 In both cases, the supremum is taken over the set of all germs of q-dimensional complex varieties V^q passing through x_0. 



Since there is only one n-dimensional complex linear subspace passing through x_0 in ^n, 
    Δ_1 (M, x_0) =Δ̃_1 (M, x_0) = D_1 (M , x_0)
 and 
    Δ_1 (, x_0) =Δ̃_1 (, x_0) = D_1 ( , x_0)
 for any ideal  in _x_0. Therefore, relating these quantities is non-trivial only if q≥ 2.

We shall also need the following result:

[Proposition 4.1 <cit.>,<cit.>]

Let  be any ideal in _x_0. For any 2 ≤ q ≤ n, D_q(, x_0) ≤Δ̃_q(, x_0).






§ PROOF OF RESULTS



We claimed in the introduction that Δ_q and Δ̃_q being simultaneously finite for ideals of germs of holomorphic functions easily follows from D'Angelo's work in <cit.>. Here is the proof:

Proof of Proposition <ref>: Let {w_1, …, w_q-1} be any non-degenerate set of linear forms in _x_0. Trivially, 

    Δ_q(, x_0)    = inf_{w_1, …, w_q-1}  Δ_1( (, w_1, …, w_q-1), x_0 ) 
       ≤gen.val_{w_1, …, w_q-1}   Δ_1( (, w_1, …, w_q-1), x_0 ) = Δ̃_q(, x_0).

Applying (<ref>) in Theorem <ref> with q-1 instead of q yields 
    gen.val_{w_1, …, w_q-1}   Δ_1( (, w_1, …, w_q-1), x_0 ) ≤gen.val_{w_1, …, w_q-1}   D ( (, w_1, …, w_q-1), x_0 )
 and 
    inf_{w_1, …, w_q-1}   D ( (, w_1, …, w_q-1), x_0 ) ≤(inf_{w_1, …, w_q-1}  Δ_1 ( (, w_1, …, w_q-1), x_0 ))^n-q+1,
 but 
    inf_{w_1, …, w_q-1}   D ( (, w_1, …, w_q-1), x_0 ) = gen.val_{w_1, …, w_q-1}   D ( (, w_1, …, w_q-1), x_0 )
 by Proposition <ref>. 

Proving that Δ_q and Δ̃_q are simultaneously finite when measured at points on the boundary of a domain requires q-positivity as well as a slightly more elaborate argument:

Proof of Proposition <ref>: Since the definition of Δ_q involves taking the infimum over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0, whereas the definition of Δ̃_q involves taking the generic value over the same sets, it is obvious that Δ_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0). We thus only have to prove the second inequality. Assume that Δ_q(b Ω, x_0) < +∞; otherwise, Δ_q(b Ω, x_0)=Δ̃_q(b Ω, x_0)=+∞, and there is nothing to prove. Now, let k be large enough so that Δ_q (M, x_0) = Δ_q (M_k, x_0), where M_k is real hypersurface defined by r_k, the polynomial with the same k-jet at x_0 as the defining function r of M; see equation (<ref>). Carry out the polarization of r_k to arrive at the ideal (U,x_0) in _x_0 defined for each unitary matrix U ∈(N). By Theorem <ref>, 
    sup_U ∈ (N)  inf_{w_1, …, w_q-1}   D( ((U,x_0), w_1, …, w_q-1), x_0 ) ≤(Δ_q(M, x_0) )^n-q.
 By Proposition <ref>, we can substitute the generic value for the infimum as follows:

    sup_U ∈ (N)  gen.val_{w_1, …, w_q-1}   D( ((U,x_0), w_1, …, w_q-1), x_0 ) ≤(Δ_q(M, x_0) )^n-q
 Since we are assuming q-positivity, Theorem <ref> and Corollary <ref> together yield 

    Δ̃_q (M, x_0)    = 2  gen.val_{w_1, …, w_q-1} sup_U ∈ (N)  Δ_1 (((U,x_0), w_1, …, w_q-1)),x_0)
       ≤ 2  gen.val_{w_1, …, w_q-1} sup_U ∈ (N)   D( ((U,x_0), w_1, …, w_q-1), x_0 ).

Since (N) is compact, Propositions <ref> and  <ref> imply that the supremum and the generic value can be exchanged, so we obtain that 
    Δ̃_q (M, x_0) ≤ 2  (Δ_q(M, x_0) )^n-q
 by combining the two previous inequalities. If M is pseudoconvex at x_0, by Proposition <ref>, M is q-positive at x_0, so the same inequality must hold. 

Our proof of the Main Theorem <ref> is motivated by a remark Catlin made on pp.147-8 of <cit.>. Catlin's remark can be paraphrased as follows: In order to compare the D'Angelo q-type with his own notion of q-type, it would be necessary to piece together one-dimensional varieties in order to get a q-dimensional one.  We start with the following lemma, which constructs a very simple q-dimensional variety containing a given curve:



Let 2 ≤ q ≤ n-1. Given a holomorphic curve Γ passing through a point x_0 ∈^n and a (q-1)-dimensional hyperplane Z passing through x_0 and satisfying that the tangent line to the curve Γ at x_0 is not contained in Z, there exists a germ of a q-dimensional cylinder C^q at x_0 that contains the curve Γ and whose tangent space at x_0 contains Z.



If x_0 is a singular point of the curve Γ, the curve could have multiple tangent lines, one for each branch of the curve. In that case, our assumption that the tangent line to the curve Γ at x_0 is not contained in Z means none of the tangent lines of Γ at x_0 are contained in Z.


Proof: Let Γ⊂ℂ^n be a holomorphic curve  (Γ =1) 
given by the equations

    f_1(z_1,z_2,...,z_n)=0



    f_2(z_1,z_2,...,z_n)=0



    ⋮



    f_n-1(z_1,z_2,...,z_n)=0


subject to the condition

    rank(∂ f_i/∂ z_j)_1 ≤ i ≤ n-1;  1≤ j ≤ n =n-1
 that holds generically, namely everywhere except at some isolated points of Γ.


Let x_0=(x_1^0, …, x_n^0). The (q-1)-hyperplane Z is given parametrically as


    x_i=Σ_j=1^q-1u_jit_j + x_i^0,   i=1,2,...,n

with the condition

    rank(u_ji)=q-1.


We have assumed that the tangent line to the curve Γ at x_0 is not contained in Z, so we can define the cylinder C^q determined by the curve Γ and the hyperplane 
Z as follows:


    C^q:={M∈ℂ^n  such that there exists  P∈Γ with line  MP parallel to  Z}.
 

By taking the coordinates for M=(z_1,...,z_n) and P=(x_1,...,x_n), we obtain


    z_i-x_i=Σ_j=1^q-1u_jit_j .


From the condition P∈Γ, we get the parametric equations of the cylinder C^q:


    f_l(-Σ_j=1^q-1u_j1t_j+z_1, ..., -Σ_j=1^q-1u_jnt_j+z_n)=0,    l=1,...,n-1.


Differentiation with respect to parameters t_1, …, t_q-1 yields


    ∂ f_l/∂ t_j=∑_i=1^n ∂ f_l/∂ x_iu_ji.


The condition that the tangent to the curve Γ at x_0 is not parallel to Z implies


    rank (∂ f_l/∂ t_j)_1 ≤ l ≤ n-1;  1≤ j ≤ q-1=q-1
 generically in the neighborhood of x_0.


Of course, we have that Γ⊂ C^q and that the tangent space of C^q at x_0 contains Z by construction. 


For each of l=1,...,n-1, the hypersurface with equation f_l(z_1,...,z_n)=0 contains the curve Γ.




Let  be an ideal of germs of holomorphic functions at x_0, then for 2 ≤ q ≤ n-1, 
    Δ̃_q(, x_0) ≤ D_q(, x_0).



Proof: We proceed as in the proof of Proposition <ref>. Let D_q(, x_0)=t < +∞, else the estimate is trivially true. Assume Δ̃_q(, x_0) >t. By definition, 
    Δ̃_q(, x_0)=gen.val_{w_1, …, w_q-1}   Δ_1( (, w_1, …, w_q-1), x_0 ),
 where the generic value is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0. Thus, there exist t'>t and a curve Γ passing through x_0 such that 
    inf_g ∈ (, w'_1, …, w'_q-1)  ord_0  Γ^* g/ord_0  Γ=t',
 for some non-degenerate set {w'_1, …, w'_q-1} of linear forms in _x_0. The variety corresponding to the non-degenerate set {w'_1, …, w'_q-1} of linear forms is a (n-q+1)-dimensional hyperplane, which we will call H. Clearly, Γ⊂ H as each w'_j has vanishing order 1 so any value starting at 2, the lower bound for Δ̃_q(b Ω, x_0), can only be achieved by a curve that sits in the zero set of {w'_1, …, w'_q-1}. Let Z be the (q-1)-dimensional hyperplane passing through x_0 that is transversal to H, i.e., _ (H ⊕ Z)=n. Since Γ⊂ H, the tangent lines to the curve Γ at x_0 (there could be several if the point is singular) are not contained in Z.  By Lemma <ref>, there exists a germ of a q-dimensional cylinder C^q_Z at x_0 that contains the curve Γ and whose tangent space at x_0 contains Z. We would like to show that Γ is one of the intersection curves along which τ((, w'_1, …, w'_q-1), C^q_Z) is measured by examining Catlin's construction in Proposition 3.1 of <cit.>. 

In the proof of Proposition 3.1 from <cit.>, Catlin removed three different sets W_1, W_2, and W_3 from G^n-q+1 in order to arrive at his generic set W̃ on which his q-type is computed. We wish to show that H ∈W̃, namely that H cannot belong to any of the three sets W_1, W_2, and W_3 taken out from G^n-q+1. First, to the germ of the variety C^q_Z, there corresponds the ideal I=(C^q_Z) in the ring _x_0 of all germs of holomorphic functions vanishing on C^q_Z. By a translation, one can suppose x_0=0 and thus denote by  the ring _x_0. Let _k denote the subring of  consisting of germs of holomorphic functions of only the first k variables.




Now, let V=C^q_Z,  and consider as in <cit.> the tangent cone of 
V at the origin, V'={z∈ℂ^n ;  f̃ (z)=0,   f∈ I}, where for any function f, 
f̃ is the homogeneous polynomial given by the leading terms in its Taylor expansion at the origin. By <cit.> the tangent cone V' can also be defined by 

    V'={lim_|z|→ 0cz/|z|; z∈ V,  c∈ℂ}

and V' has dimension q since V has dimension q. The curve Γ in the (n-q+1)-dimensional hyperplane H has as its tangent cone at the origin a union of finitely many lines L_1, … , L_s through the origin that are all contained in H. 

We take advantage of the fact that the q-dimensional variety C^q_Z has a particularly simple structure by construction.  
Since the hyperplane H is given by a non-degenerate set w'_1, … , w'_q-1 of linear forms in , by making a linear change of variables, 

we can assume that the equations of the (n-q+1)-dimensional hyperplane H are z_1=0, … , z_q-1=0. Then the equations of the curve Γ have the following form (see Lemma <ref>):


    f_1=z_1=0,… , f_q-1=z_q-1=0, f_q=f_q(z_1,… ,z_q-1, z_q,… , z_n)=0,
 
    … , f_n-1=f_n-1(z_1,… ,z_q-1, z_q, … , z_n)=0.


By choosing the parametric equations of the hyperplane Z to be of the form

    z_1=t_1, … ,z_q-1=t_q-1, z_q=0, …, z_n=0,

we get for the parametric equations of the cylinder C^q_Z (see the proof of Lemma <ref>):


    z_1-t_1=0, … , z_q-1-t_q-1=0, f_q(0,… ,0, z_q, … , z_n)=0,


    … , f_n-1(0, … ,0, z_q, … , z_n)=0.


Therefore, the equations for the cylinder C^q_Z are given by


    f_q=f_q(0,… ,0, z_q, … , z_n)=0,… , f_n-1=f_n-1(0, … ,0, z_q, … , z_n)=0,

which are the equations of the curve Γ in the hyperplane H.

Let us denote by I_1 the ideal generated by (f_q, …, f_n-1) in the ring 
 and by J the ideal generated by (f_q, …, f_n-1) in the ring 
_q,… ,n, the ring of germs of holomorphic functions at 0 in the hyperplane H≅ℂ^n-q+1. We have the relations


    I=rad (I_1), _q,… ,n⊂,  J⊂ I.

Let _k_q,… ,n denote the subring of germs of holomorphic functions of 
only the variables z_q, … , z_k, where k=q+1, … , n. 

Using Gunning's Local Parametrization Theorem from p.15 of <cit.>, since the dimension of Γ in H is 1, one can choose functions 
g_k,  k=q+1, … , n and linear coordinates (z_q, z_q+1, … , z_n) such that: (1) g_k∈ J∩_k-1_q,… ,n [z_k], (2) g_k is a Weierstrass polynomial in z_k of degree m_k, and (3) _q_q,… ,n∩ J=0.

Moreover, at the last step when g_q+1 is chosen, by a linear change of variables in z_q and z_q+1, we can suppose that the (n-q)-dimensional hyperplane given by the equation z_q=0 in the hyperplane H does not contain the lines L_1, … , L_s. This can be seen as follows: Take the projective space 
ℙ^n-q associated to the hyperplane H, and let π :H→ℙ^n-q be the canonical projection. The lines L_1, … , L_s will define s points in the projective space ℙ^n-q and we can choose a hyperplane Y of dimension (n-q-1), which does not contain any of these points. π^-1(Y), the inverse image of Y in H, then does not contain the lines L_1, … , L_s. Now, the orthogonal line to π^-1(Y) in H is the z_q-axis as _q_q,… ,n∩ J=0 by condition (3) above.

We remark that _q-1∩ I=0. Indeed, if h(z_1, … , z_q-1)∈ I, then we have for some j that h^j ∈ I_1, so 
h^j=a_qf_q+ … + a_n-1f_n-1, where a_q, …, a_n-1 are germs of holomorphic functions at 0. Setting in this relation z_q=0, …, z_n=0 yields
h^j=0, hence h=0. Since _q,… ,n⊂,  J⊂ I, we obtain that the functions g_k,   k=q+1, … ,n satisfy the conditions:  
(1) g_k∈ I∩_k-1 [z_k], (2) g_k is a Weierstrass polynomial in z_k of degree m_k. Furthermore, from the conditions _q-1∩ I=0 and _q_q,… ,n∩ J=0, it follows that (3) _q∩ I=0.

From the existence of g_k,   k=q+1, … ,n, by using some facts about integral extensions (see <cit.>), one can conclude that there exist Weierstrass polynomials p_k∈ I∩_q [z_k], k=q+1,…, n. By choosing each function p_k to be one of the minimal degree with this property, then since I equals its radical, one may suppose that each p_k is a product of p_k,j, j=1,…, N_k of distinct irreducible in _q [z_k]. It follows that the discriminant d_k of such a product is in the ring _q and not equal to the zero-element (see <cit.>). 

Since the q-dimensional variety V=C^q_Z has a particularly simple structure by construction, it follows from the second description of V' (expression (<ref>)) that the tangent cone of V is given by the finite union of q-dimensional hyperplanes Z× L_1, … , Z× L_s. 

Catlin defined in <cit.> another conic variety X'⊂ V' by 


    X'={z∈ V';   z_qg̃_q∏_q+1^n d̃_k(z)=0},

where  g̃_q=1 in our case since the cylinder variety V=C^q_Z is pure q-dimensional. It follows that X' is a conic subvariety of dimension q-1 in ℂ^n, which gives rise to a projective subvariety X̃ in ℙ^n-1 of dimension q-2. If S ∈ G^n-q+1 and S̃ denotes the corresponding projective hyperplane of dimension n-q in ℙ^n-1, then
the set W_1 consists of all (n-q+1)-dimensional complex hyperplanes  S such that S̃∩X̃≠∅. We have to show that H̃∩X̃=∅. Firstly, we have V'∩ H=∪_j=1^s L_j, hence Ṽ∩H̃ has s points corresponding to the lines L_1, … , L_s. For the intersection X'∩ H, we have to inspect the equation of X' when z_1=0, … , z_q-1=0. We obtain z_q∏_q+1^n d̃_k(0,0, … ,0, z_q)=0 whose only solution is z_q=0 because d̃_k is a homogeneous polynomial for every k. The (n-q)-dimensional hyperplane of H given by this solution does not contain any of the lines L_1, … , L_s as it is π^-1(Y) described above. It follows that H̃∩X̃=∅, i.e. H∉ W_1.







Secondly, to ensure the intersection C^q_Z ∩ S for S ∈ G^n-q+1 behaves well, a good notion of transversality has to apply. 
For every S ∈ G^n-q+1, there corresponds a projective plane S̃ of dimension n-q in ^n-1. Generically, C̃^q_Z ∩S̃ consists of finitely many points z̃^1, …, z̃^D with transverse intersections, meaning that each z̃^i is a smooth point of C̃^q_Z such that the tangent spaces satisfy T_z̃^iC̃^q_Z ∩ T_z̃^iS̃ = 0 for i=1, …, D. In <cit.> W_2 is the subset of G^n-q+1 where this generic behavior does not take place. As we have seen Ṽ∩H̃ has s points corresponding to the lines L_1, … , L_s. It follows that D=s, and we denote by 
z̃^i the point corresponding to L_i. Since L_i is a line in Z× L_i and in H, we have that T_z̃^iṼ=Z× L_i/L_i≅ Z and T_z̃^iH̃=H/L_i. Z∩ H=0, however. It follows that T_z̃^iṼ∩ T_z̃^iH̃ = 0 for all i=1,...,s, i.e. H∉ W_2.










Finally, suppose L∈ G^n-q+1 is defined by L={z; ∑_j=1^na^i_jz_j=0,   i=1, … , q-1}. Let W_3={L;  [a^i_j]_1≤ i,  j≤ q-1=0}. Since the equations of H are z_1=0, ..., z_q-1=0, we have [a^i_j]_1≤ i,  j≤ q-1=1, and 
we conclude H ∉W_3.

We have shown that H ∈W̃, so the curve Γ enters into the computation of τ((, w'_1, …, w'_q-1), C^q_Z). Therefore, 
    D_q(, x_0) ≥inf_g ∈ (, w'_1, …, w'_q-1)  ord_0  Γ^* g/ord_0  Γ=t' > t = D_q(, x_0),
 giving us the needed contradiction. 





Let Ω in ^n be a domain with  boundary. Let x_0 ∈ b Ω be a point on the boundary of the domain, and let 2 ≤ q ≤ n-1. Then 
    Δ̃_q(b Ω, x_0)≤ D_q(b Ω, x_0).



Proof: The proof is very similar to that of the previous result, Propostion <ref>. Let D_q(b Ω, x_0)=t < +∞, else the estimate is trivially true. Assume Δ̃_q(b Ω, x_0) >t. By definition, 
    Δ̃_q(b Ω, x_0)=gen.val_{w_1, …, w_q-1}   Δ_1( ((b Ω), w_1, …, w_q-1), x_0 ),
 where the generic value is taken over all non-degenerate sets {w_1, …, w_q-1} of linear forms in _x_0 and 
    Δ_1( ((b Ω), w_1, …, w_q-1), x_0 ) = sup_φ∈(n,x_0)  inf_g ∈((b Ω), w_1, …, w_q-1)  ord_0  φ^* g/ord_0  φ.
 Therefore, there exist t'>t and a curve Γ passing through x_0 such that 
    inf_g ∈ ((b Ω), w_1, …, w_q-1)  ord_0  Γ^* g/ord_0  Γ=t',
 for some non-degenerate set {w'_1, …, w'_q-1} of linear forms in _x_0. We will call H the variety corresponding to the non-degenerate set {w'_1, …, w'_q-1} of linear forms, which is a (n-q+1)-dimensional hyperplane. Clearly, Γ⊂ H as each w'_j has vanishing order 1 so any value starting at 2, the lower bound for Δ̃_q(b Ω, x_0), can only be achieved by a curve that sits in the zero set of {w'_1, …, w'_q-1}. Therefore, inf_g ∈ ((b Ω), w_1, …, w_q-1) is achieved for g=r, the defining function of the domain Ω. Let Z be the (q-1)-dimensional hyperplane passing through x_0 that is transversal to H, i.e., _ (H ⊕ Z)=n. Since Γ⊂ H, the tangent line to the curve Γ at x_0 is not contained in Z.  By Lemma <ref>, there exists a germ of a q-dimensional cylinder C^q_Z at x_0 that contains the curve Γ and whose tangent space at x_0 contains Z. By the same analysis as in the proof of Propostion <ref>, H ∈W̃, so the curve Γ enters into the computation of τ(C^q_Z, x_0). Thus, 
    D_q(b Ω, x_0) ≥ord_0  Γ^* r/ord_0  Γ=t' > t = D_q(, x_0),
 and we have obtained the contradiction we sought. 
  


Proof of Theorem <ref>: First, we prove part (i). For q=n Proposition <ref> along with Theorem  <ref> yield that Δ_n(,x_0)=Δ̃_n(,x_0)=D_n(,x_0). As a result, the equality only needs to be proven for 2 ≤ q ≤ n-1. The inequality D_q(, x_0) ≤Δ̃_q(, x_0) is a consequence of Theorem <ref>, while Δ̃_q(, x_0) ≤ D_q(, x_0) follows from Proposition <ref>.

To prove part (ii), we note that D_q(b Ω, x_0) ≤Δ̃_q(b Ω, x_0) is a consequence of Theorem <ref>, while the reverse inequality follows from Proposition <ref>.  

Proof of Corollary <ref>: (i) We combine Proposition <ref> with Theorem <ref> part (i).

(ii) Proposition <ref> and Theorem <ref> part (ii) together give the result. 

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