Relative genericity of controllablity and stabilizability for differential-algebraic systems

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices ( E , A , B ) ∈ R (cid:2) × × R (cid:2) × n × R (cid:2) × m , while for relative genericity we allow the restricted set (cid:3) ≤ r (cid:2), n , m := { ( E , A , B ) ∈ R (cid:2) × n × R (cid:2) × n × R (cid:2) × m (cid:2)(cid:2) rk R E ≤ r } , where r ∈ N . Our main results are characterizations of generic controllability and generic stabilizability in (cid:3) ≤ r (cid:2), n , m in terms of the numbers (cid:2), n , m , r .

· p×q an operator norm on R p×q (E, A, B) := max E ×n , A ×n , B ×m , the norm on R ×n × R ×n × R ×m f −1 (A) := {x ∈ X : f (x) ∈ A}, the preimage of the set A ⊆ Y under the function f : X → Y R[x 1 , . . . , x n ] : = k=0 a k x ν k, 1 1 · · · x ν k,n n ∈ N, a k ∈ R, ν k, j ∈ N , the ring of (real) polynomials in n indeterminants

Introduction
In a predecessor [10] of the present note, we characterized genericity of controllability and stabilizability of differential algebraic systems (DAEs) described by the equation where (E, A, B) ∈ ,n,m := R ×n × R ×n × R ×m and , n, m ∈ N * .
The notion of genericity is defined as follows. such that V is the locus of their zeros, i. e.
An algebraic variety V is called proper if V R n . The set of all algebraic varieties in R n is denoted as V (R n ) := V ⊆ R n ∃ q 1 (·), . . . , q k (·) ∈ R[x 1 , . . . , x n ] : and the set of all proper algebraic varieties as A set S ⊆ R n is called generic, if there exist a proper algebraic variety V ∈ V prop n (R) so that S c ⊆ V. If the algebraic variety V is known, then we call S generic with respect to (w.r.t.) V.
When the concept of genericity is applied to differential algebraic systems (E, A, B) ∈ ,n,m , a drawback is that the set ,n,m is too "large". By this we mean that if = n, then in each arbitrarily small neighbourhood of (E, A, B) ∈ ,n,m there is some E ∈ R ×n with full rank, and this yields that (E , A, B) is an ordinary differential equations. To be more precise, the set ODE n,m = (E, A, B) ∈ n,n,m E ∈ Gl(R n ) is the nonempty complement of the preimage of zero under the polynomial and therefore a generic set. Each system (1) that is described by a triple (E, A, B) ∈ ODE n,m corresponds to the ordinary differential equatioṅ For this case, Lee and Markus [12] proved that that the set of all controllable systems is open and dense w.r.t. the Euclidean topology, and Wonham [18,Thm. 1.3] showed in the first edition of his monograph that the set of all controllable systems is generic. We are aiming at reference sets where E does not have full rank, these are the systems (E, A, B) of (1) belonging to However, the question as to whether controllability and stabilizability holds for one of these system classes is not well posed. If ≤r ,n,m and =r ,n,m were (affine) subspaces, then we could identify them with a real coordinate space and study whether the sets of matrix triples (E, A, B) ∈ ≤r ,n,m and (E, A, B) ∈ =r ,n,m whose corresponding DAE is controllable are generic subsets in the sense of Definition 1.1. Unfortunately, these sets are (in general) not a subspace and hence Definition 1.1 is not readily applicable. To resolve this, we propose to study relative genericity of controllability and stabilizability -a concept recently introduced by Kirchhoff [11]. and Relative genericity of controllability has already been shown for linear port Hamiltonian systems without dissipation, see [11]. In the following, we relate relative genericity to other concepts of genericity known in the literature.
First of all, recall what "generic" means in a topological context. In a Baire space (X , O), a set S ⊆ X is called generic if, and only if, there are open, dense sets D i ∈ O, i ∈ N, so that i∈N D i ⊆ S, see [9, p. 45]. A stronger formulation of this, which can be extended to arbitrary topological spaces, is that a set is generic if, and only if, it contains some open and dense set. This definition was used e. g. by Banaszuk and Przyłuski [3] to show that the systems (1) with (E, A, B) ∈ ≤r ,n,m are generically controllable and stabilizable. Unfortunately, their notions of controllability and stabilizability are different to the standard concepts treated by the control theory community, see for example the survey by Berger and Reis [4,Cor. 4.3].
The two definitions of genericity given in the latter paragraph are purely topological. However, the R n is not only intrinsically a Baire space (with the Euclidean topology) but also a complete measure space (with the Lebesgue measure). Therefore, we should also consider genericity from a measure theoretic point of view. If (X , A , μ) is a complete measure space (or μ just an outer measure), then we could call some set S ⊆ X generic if, and only if, there is some μ-nullset N ∈ A so that X \ S ⊆ N . This concept is e. g. used by Vovk [17]. A reference from mathematical systems theory might be the survey article by Hunt and Kaloshin [9], where prevalence is considered: a subset S of the complete metric Abelian group X is prevalent if, and only if, its complement is a shy set, i. e. a nullset with respect to a nontrivial measure on X so that all its translations are nullsets, too. These properties are essentially the properties of the Lebesgue-measue on R n , i. e. Lebesgue-almost-everywhere implies prevalence. It can be shown that the converse holds true. [9, Proposition 2.5] The concept of prevalence has been extended to relative prevalence on completely metrizable convex subsets of X , see [2]. Unfortunately, neither of our reference sets =r ,n,m and ≤r ,n,m is convex. Furthermore, these measure-theoretic concepts do largely not consider the topological properties of generic sets. Therefore, we would like to see a concept that includes both the topological and the measure theoretic point of view. Intuitively, such a definition is of the form where λ n denotes the n-dimensional Lebesgue measure. Genericity as in Definition 1.1 does fulfil this requirement: each proper algebraic variety is a nowhere dense Lebesgue nullset, see [6, p. 240]. Therefore, Wonham's definition is stronger than the intuitive definition (5). Considering any co-countable set, we see that it is strictly stronger. The drawback that some sets such as R \ N should intuitively be generic can be resolved when considering analytic everywhere defined functions instead of polynomials; the resulting concept remains strictly stronger than the intuitive definition (5) and is strictly weaker than Wonham's definition. A problem that occurs in generalizing this concept to arbitrary subsets of R n is that not each subset is intrinsically a measure space with a nontrivial measure. We choose to ignore this fact and try to define a genericity that is topological with respect to the Euclidean relative topology and measure theoretic for special submanifolds. When sticking to the algebraic varieties used in Wonham's definition, this leads to relative genericity as in Definition 1.2. The paper is organized as follows. Section 2 is devoted to the properties of relative generic sets and some illustrating examples. We give a topological characterization of relative generic sets and briefly discuss some measure theoretic properties. The main results, that are the characterizations of relative genericity of controllability and stabilizability for sets with respect to the reference set ≤r ,n,m in terms of the dimensions , n, m and the maximal rank r , are given in Sections 3 and 4 , resp. The proofs in these sections rely heavily on the well-known algebraic characterizations of controllability and stabilizability (see Propositions 3.1 and 4.1 ) and on subtle and involved technicalities, which we have relegated to "Appendix A".

Relative generic sets
In the present section, we present elementary properties of relative generic sets. In the introduction, we claimed that relative genericity as in Definition 1.2 can be viewed from a topological and, provided that the reference set admits some "structure", from a measure theoretic point of view. Since we consider subsets of the n-dimensional Euclidean space, we equip each reference set with its Euclidean relative topology. Additionally to the Euclidean relative topology we consider the coarser Zariski relative topology. Recall [15, pp. 50] that the Zariski topology is defined by the property that all closed sets are the algebraic varieties. In terms of these topologies, we can characterize relative generic sets as follows. Proof Let S be relative generic in V . By Definition 1.2, there is some algebraic variety Conversely, if there is some set O ⊆ S that is open w.r.t. the relative Zariski topology on V and dense w.r.t. the relative Euclidean topology on V , then O is necessarily of the form V c ∩ V for some algebraic variety V. Unless V = ∅, Euclidean density of O yields that O is nonempty or, equivalently, V a proper algebraic variety. If V = ∅, however, then O = V ∩ V for all V ∈ V prop (R n ) and hence we can w.l.o.g. assume that V ∈ V prop (R n ). Therefore, we conclude that there is indeed some proper algebraic variety V so that the properties (RG1) and (RG2) are fullfilled. Thus, S is relative generic in V .
This characterization yields that genericity and relative genericity coincide for the reference set R n . In this case Zariski open and Zariski dense sets are Euclidean dense.
In [11] we have already mentioned that we need to impose some additional condition on relative generic sets beside containing a nonempty Zariski open set; otherwise we can show that each co-finite subset of a discrete reference set is relative generic, which is not consistent with the intuitive meaning of genericity. Since the Euclidean topology is the intrinsic topology on R n , we would like to see density with respect to the Euclidean relative topology on the reference set. Here we ask as to whether it suffices to require density in (RG2) in the relative Zariski topology. This, however, is generally not the case. The following proposition contains a small collection of properties of relative generic sets. 1 yields that if S 1 and S 2 are relative generic in V , then their intersection and union is likewise relative generic in V . Conversely, if S 1 ∩ S 2 is relative generic in V , then (c) yields with the trivial inclusions S 1 ∩ S 2 ⊆ S 1 and S 1 ∩ S 2 ⊆ S 2 that both S 1 and S 2 are relative generic in V .
(e) is consequence of the properties of the product topology and Lemma 2.1.
We It remains to prove (g): Assume that both S 1 and S c 1 are relative generic in V . By Lemma 2.1, we conclude that there are two disjoint Euclidean open and dense sets in the relative topology on V . This yields that either one of these sets is empty and therefore V = ∅, a contradiction to the assumption. Therefore there is no non-trivial partition of a given reference set in relative generic sets.
A first result on the reference sets (E, A, B) ∈ ≤r ,n,m and (E, A, B) ∈ =r ,n,m is the following.

Lemma 2.4 =r
,n,m is relative generic in ≤r ,n,m for each , n, m ∈ N * , r ∈ N.
Proof Let M 1 , . . . , M q be all minors of order r with respect to R ×n and consider the mappings Then the identity holds true. In view of Definition 1.2, it remains to prove that Let (E, A, B) ∈ ≤r ,n,m and ε > 0. Then and the perturbation matrix Therefore, =r ,n,m is a dense subset of ≤r ,n,m .
As a consequence of Proposition 2.3(v) and Lemma 2.4 we derive the following corollary that ≤r ,n,m and =r ,n,m are interchangeable reference sets.

Corollary 2.5 A set S ⊆ ,n,m is relative generic in ≤r
,n,m if, and only if, S ⊆ ,n,m is relative generic in =r ,n,m .
The proofs of our main Theorems 3.2 and 4.2 depend crucially on the relative genericity of some rank properties. This is the content of the following proposition. Since the proofs are rather lengthy, we only state the result and relegate the proofs to "Appendix A". Proposition 2.6 Let r ∈ {0, . . . , min { , n}} be arbitrary. Then the sets An interesting observation about Proposition 2.6 is that only the sets S r (i) and S r In the remainder of this section we interprete relative generic sets from a measure theoretic point of view for special submanifolds. In the next proposition it is shown that these special submanifolds are at least those, who admit a countable induced atlas. We recall the necessary definitions.
The pair (U , ϕ) is a local submanifold chart. Any collection (U i , ϕ i ) i∈I of local submanifold charts with i∈I U i ⊃ M is a submanifold atlas. This atlas induces the family (U i ∩ M, ϕ i U i ∩M ) i∈I , which is a atlas of local manifold charts for M, the induced atlas.

Proposition 2.8 Let V be an analytic submanifold of R n , n ∈ N * , so that an induced atlas is countable and S is relative generic in V . Then S c ∩ V is a nullset with respect to the Riemann-Lebesgue measure on V .
Proof Since V is a submanifold of the Riemannian maniold R n (with the standard scalar product as Riemannian metric), it is a Riemannian manifold itself and allows therefore the definition of a Riemann-Lebesgue measure. We show that each algebraic variety V ∈ V prop (R n ) which fullfills the condition (RG2) is a nullset with respect to the Riemann-Lebesgue measure on V . First, we recall a characterization of nullsets with respect to the Riemann-Lebesgue measure on a Riemannian manifold M: A set A ⊆ M is a nullset if, and only if, for each chart (ϕ, U ) of a countable atlas of M the set ϕ(A ∩ U ) is a Lebesgue nullset. Since we require for the induced atlas of V to be countable, it suffices to show that the image of S c ∩ V under the charts of this atlas is a nullset. These charts however are analytic diffeomorphisms and therefore the image of S c ∩ V under those the preimage of zero under analytic mappings. The requirement that S c ∩ V is nowhere dense implies that these analytic mappings are nonzero. Federer [6, p. 240] proved that the preimage of zero under nonzero analytic functions is a Lebesgue nullset. Therefore, we conclude that S c ∩ V is indeed a nullset with respect to the Riemann-Lebesgue measure on V .
We would like to relate the previous proposition to our reference set ≤r ,n,m . This set is an algebraic variety. Assume that the ideal of polynomials which vanish on ≤r ,n,m is generated by p 1 , . . . , p k . Recall that (E, A, B) ∈ ≤r ,n,m is called nonsingular [15, p. 97] if the rank of the Fréchet derivative of ( p 1 , . . . , p k ) at (E, A, B) is maximal. Using the implicit function theorem as in [16, pp. 88-89] it can be shown that each nonsingular (E, A, B) has an open neighbourhood U so that V = ≤r ,n,m ∩ U is an analytic submanifold with a single chart. Therefore, the induced atlas is especially countable and we can apply Proposition 2.8 to find that relative genericity in ≤r ,n,m yields Riemann-Lebesgue-almost-surely near nonsingular points. It can be shown that the nonsingular points are contained in =r ,n,m . This implies that the previous statement holds true for =r ,n,m , too. Further research in this direction, however, is not part of the present note. The purpose of the previous discussion was to show how a measure theoretic interpretation of relative genericity could be attempted.

Controllability
In this section, we first recall the well-known definitions and characterizations of freely initializable, impulse controllable, behavioural controllable, completely controllable and strongly controllable systems. The algebraic criteria are then used to derive necessary and sufficient conditions for the sets of all matrix triples so that the corresponding DAE system (1) is controllable (in one of the five mentioned ways) is relative generic in the reference set ≤r ,n,m . We fix, for the remainder of this note, , n, m ∈ N * and r ∈ N.
The following controllability definitions rely on the notion of behaviour used for (1) and defined as A, B) ∈ ,n,m the following controllability definitions associated to the system (1) are algebraically characterized as follows:

Proposition 3.1 For any (E,
Moreover, the followoing equivalences are valid: A, B) freely initial. and beh. contr.

Proof
in conjunction with the algebraic characterization of behavioural controllability. It remains to show that "completely controllability ⇐ free initializability and behavioural controllability". The presuppositions give Now the claim follows from the algebraic characterization of complete controllability.
Assertion (b) can be shown similarly to Assertion (a) by replacing [E, B] with [E, AZ, B] for some Z ∈ R n×(n−rk E) with im Z = ker E. This is omitted.
We are now ready to prove our main result, that is the characterization of relative genericity of the different controllability concepts in terms of the system dimensions.
To this end we introduce the notation where 'controllable' stands for one of the controllability concepts. Using these algebraic characterizations, we find necessary and sufficient conditions on , n, m ∈ N * and r ∈ N so that S controllable is relative generic in ≤r ,n,m . As we already discussed after Proposition 2.6, it is not very surprising that we recover, unless r = 0, for impulse controllability, behavioural controllability and strong controllability, the necessary and sufficient conditions for the unrestrained case, see [10,Theorem 2.3].
Recall that by Corollary 2.5 the following statements do also hold for the reference set =r ,n,m . Theorem 3.2 The following equivalences hold: else.
Therefore, and since ∅ is not relative generic in =r ,n,m = ∅, we have that S freely init. is relative generic in =r ,n,m if, and only if, min { , r + m} = min { , r + n + m} or, equivalently, ≤ r + m. else.
Similar to (a), we conclude that S imp. contr. is relative generic in =r ,n,m if, and only if, min { , n + m} = min { , r + n + m} or, equivalently, ≤ n + m or r = 0.
holds and therefore S beh. contr. is relative generic in =r ,n,m . Therefore, Proposition 2.3(b) and (c) give that S compl. contr. is relative generic in =r ,n,m if, and only if, both S freely init. and S beh. contr. are relative generic in =r ,n,m . Since we have already proved the statements (a) and (c) of the present theorem, it remains to prove holds. Analogously to the proof of statement (d) of the present theorem we conclude that the former set is relative generic in =r ,n,m if, and only if, both latter sets are relative generic in =r ,n,m and otherwise S c str.contr. is relative generic in =r ,n,m . Using the results of the statements (b) and (c) of the present theorem, it remains to verify which is straightforward. This completes the proof of the proposition.

Stabilizability
In the present section we recall the well-known definitions and characterizations of completely stabilizable, strongly stabilizable and behavioral stabilizable systems. Using these algebraic characterizations in combination with some results from the "Appendix A", we derive necessary and sufficient criteria in terms of systems dimensions when stabilizability of (1) holds relative generically in ≤r ,n,m . Proposition 4.1 Let , n, m ∈ N * . We use, as for controllability, the abbreviation S stabilizable := (E, A, B) ∈ ,n,m (1) stabilizable .
Considering the concepts of completely stabilizable, strongly stabilizable and behavioural stabilizable systems, the following characterizations can be found Similar as for controllability, Proposition 2.6 in conjunction with the algebraic criteria from Proposition 4.1 give necessary and sufficient conditions on , n, m and r so that S stabilizable is relative generic in ≤r ,n,m . Unless r = 0, we recover, as discussed for controllability, the necessary and sufficient conditions for the unrestrained case, see [

Proof
Since ∅ is not relative generic, we conclude that relative genericity of S compl. stab. yields ≤ r + m. By Propositions 4.1 and 2.6 give Therefore, Proposition 2.3(b) and (c) ensure that S compl. stab. is relative generic in =r ,n,m if, and only if, S (vi) is relative generic in =r ,n,m . Now Proposition 2.6 gives that relative genericity of S compl. stab. implies This condition is equivalent to the equivalent conditions ≤ r + m ∧ ¬ ( = n + m ∧ r = n) ⇐⇒ ≤ min {r , n − 1} + m. (8) It remains to prove the converse implication. Assume that ≤ min {r , n − 1} + m holds. Then (8) yields ≤ r + m and we conclude (7). Since (8) yields = n + m ∨ r = 0 , we may apply Proposition 2.6 to conclude that S (vi) is relative generic in =r ,n,m . Finally, me may apply, Proposition 2.3(b) and (c) to (7) to conclude that S compl. stab. is relative generic in =r ,n,m .
Step (a). Then a similar reasoning gives that S str. stab. is relative generic in =r ,n,m if, and only if, This condition is equivalent to < n + m + min {r , 1} . (c) From Proposition 2.6 we know that S (iii) is relative generic in =r ,n,m . By Proposition 2.3(b) and (c), S beh. stab. is relative generic in =r ,n,m if, and only if, the intersection S beh. stab. ∩ S (iii) is relative generic in =r ,n,m . In view of Proposition 4.1, the equality holds. Therefore, Proposition 2.6 implies that S (vi) ∩ S (iii) is relative generic in =r ,n,m if, and only if, = n + m or r = 0. This completes the proof of the theorem.

A Appendix
In this section we prove the results that form the basis of the proofs of Theorem 3.2 and Theorem 4.2. We use the well-known minors and the resultant of two polynomials, and recall their definition and properties.
In the next lemma we exploit Lemma A.4 to prove some rank inequalities. (E, A, B) ∈ ,n,m with Z ∈ R n×(n−rk E) such that im Z = ker E we have

Lemma A.5 For any
Proof The first inequality is immediate.
We prove the second inequality. Let v 1 , . . . , v K be linear independent columns of [E, AZ, B] so that v 1 , . . . , v k are columns of [E, B] and v k+1 , . . . , v K are columns of AZ. Then Furthermore, We show that the family is linearly independent over quotient field R(x), this would complete the proof of the second inequality. Equivalently, we may show that V is linearly independent over the ring Seeking a contradiction, suppose that Writing comparing coefficients in (11) and invoking that v i comes with x and w i with x 0 yields in particular and thus linear independence of v 1 , . . . , v K gives p 1,μ = . . . = p K ,μ = 0. This contradicts (12) and completes the proof of the second inequality. It remains to prove the third inequality. Let k := rk R(x) [x E − A, B]. If k = 0, then the third inequality is clear. Hence we restrict ourselves to the case k > 0. Then there exists a minor M of order k w.r.t. Let M 1 , . . . , M q be all minors of order k w.r.t. R k×2k . By Lemma A.4, we find some This proves the third inequality and completes the proof of the lemma.
Remark A. 6 An alternative proof of the third inequality in Lemma A.5 can be given by exploiting the quasi Kronecker form. Since the rank of a matrix is invariant under multiplication with regular matrices, it suffices to show for some S ∈ GL (R) and T ∈ GL n (R). Applying the quasi Kronecker form from [5,Thm. 2.6] yields the existence of some S ∈ GL (R) and T ∈ GL n (R) such that for some matrices of appropriate sizes with some specific properties. We can then exploit the block-diagonal structure of the transformed matrices S E T and S AT and the properties of the polynomial matrices x E P − A P , x E R − A R and x E Q − A Q to prove the asserted inequality. An advantage of the present proof of Lemma A.5 -not invoking the quasi Kronecker form -is that it is "straightforward" to generalize it for arbitrary polynomial block matrices over integral domains. However, the calculations are involved and since we do not need the general case for the present note, we omit the details.
The main result of this section is the proof of Proposition 2.6, where we show that the rank of several block matrices of (E, A, B) ∈ ,n,m is generically "full" on =r ,n,m Proof of Proposition 2. 6 Relative genericity of S r (i) and S r (ii) can be proven in an analogous manner to the proof of Lemma 2.4 and therefore omitted.
We show that that S (iii) is relative generic in =r ,n,m . If the inclusion then it is easy to see that the set is generic (the proof is analogous to the proof of [10,Proposition B.3]). Therefore, Proposition 2.3(d) yields that S (iii) is relative generic in =r ,n,m . By Proposition 2.3(ii), S (iii) is relative generic in =r ,n,m . It remains to show the inclusion (14). Equivalently, we prove that every (E, A, B) ∈ ,n,m satisfies the implication Let Therefore, we have equality in (15) and thus (E, A, B) ∈ S (iii) . This shows that the inclusion S (iii) ⊆ S (iii) holds indeed true. We show that S (iv) is relative generic in =r ,n,m and distinguish two cases. Case r = n: In this case the identity =n ,n,m ∩ S (iv) = =n ,n,m ∩ S r holds true. Since S r (ii) is relative generic in =r ,n,m , we conclude that S (iv) is relative generic in ≤r ,n,m . Case n > r : We proceed in steps.
Step (iv).1: We construct, for ρ ∈ {0, . . . , r }, a mapping T ρ that applies the first ρ steps of the Gaussian elimination to an × n-matrix. Let T 0 be the identity mapping on R ×n and For each i ∈ r , we define the set dom T i := E ∈ R ×n E i,i = 0 and the mappings The zeros in the last −i rows of T i (E) for E ∈ dom T i appear in the i-th column. The mapping T i , i ∈ r , is the i-th step of the Gaussian algorithm without interchanging the rows and without normalization. Let, for each j, k ∈ N * , π j,k : be the projection onto the entry ( j, k) of a matrix. Using this notation, the mappings τ i can be viewed as rational matrices, i. e.
Concatenation of the mappings T i yields, for ρ ∈ {0, . . . , r }, the mappings By definition of the mappings T i , i ∈ ρ, and dom T ρ we have Especially, we conclude Step (iv).2: We prove by induction on ρ ∈ {0, . . . , r } that T ρ can be viewed as a rational matrix, i. e.
Hence, we get with the induction assumption π ρ+1,ρ+1 • T ρ ∈ R(x 1 , . . . , ...,x n ) . Therefore, the mapping τ ρ+1 • T ρ can also be viewed as a rational matrix. With and the induction assumption, we conclude that the mapping T ρ+1 can be written as the product of rational matrices. Therefore, (22) holds true for T ρ+1 .
By our choice of z i r + j , i, j ∈ n − r , the family z i (E) i∈n−r ∈ R n n−r is linearely independent for each E ∈ dom T r . Moreover, since T r can be written as the multiplication with the regular matrices τ 0 , τ 1 • T 0 , . . . , τ r • T r −1 , we have ∀ E ∈ dom T ρ : ker E = ker(T r E).
polynomial for each i ∈ q. As we discussed in Step (iv).5, dom T is the complement of a proper algebraic variety. Therefore, the set is a proper algebraic variety with S (iv) = V c , and hence ≤r ,n,m \ S (iv) ⊆ V. It remains to show that ≤r ,n,m ∩ V c = ≤r ,n,m ∩ S (iv) is dense in ≤r ,n,m . We choose the norm · : ,n,m → R ≥0 , (E, A, B) → max E ×n , A ×n , B ×m . (29) Let ε > 0 and (E, A, B) ∈ ≤r ,n,m . Since dom T r is relative generic in V , there is some H ∈ dom T r ∩ V r so that E − H ×n < ε.
With this matrix H , we consider This proves that S (iv) ∩ ≤r ,n,m is dense in ≤r ,n,m . Since S (iv) is the complement of a proper algebraic variety, it is relative generic in ≤r ,n,m .
Step (iv).8: It is easy to see that the inclusion S (iv) ∩ ≤r ,n,m ⊆ S (iv) ∩ ≤r ,n,m holds true. Therefore, we conclude with Proposition 2.3(ii) that S (iv) is relative generic in ≤r ,n,m . We show that S (v) is relative generic in =r ,n,m . We first show, that the condidition = n + m or r = 0 is indeed necessary for S (v) to be relative generic in =r ,n,m . Let = n + m and r > 0. Analogeous to the proof of Lemma 2.4 it is easy to see that the set S = (E, A, B) ∈ ,n,m deg det[x E − A, B] = r is relative generic in =r ,n,m . The fundamental theorem of algebra yields that S ⊆ S c (v) . Hence Proposition 2.3(ii) yields that S c (v) is relative generic in =r ,n,m . Since nowhere dense sets cannot contain an open and dense set, we conclude that especially S (v) cannot be relative generic in =r ,n,m . It remains to show sufficiency. In the case r = 0 the statement is equivalent to the statement of (ii). Hence we can assume r > 0 and = n + m. We proceed in steps.
Step V( E) so that A − A ×n < ε and B − B ×m < ε. Since ( E, A, B) ∈ V c ∩ =r ,n,m , we conclude that V c ∩ =r ,n,m is dense in =r ,n,m .
Step (v).2.2: The remaining case n + m < can be treated analogously to Step(v).2.1 and is therefore omitted.
Step (v).3: We have shown that there is some algebraic variety V ∈ V (R N ) so that S c (v) ∩ =r ,n,m ⊆ V and V c ∩ =r ,n,m is dense in =r ,n,m . Since =r ,n,m = ∅, we conclude that V is indeed a proper algebraic variety. Therefore, the conditions of Definition 1.2 are fulfilled and S (v) is relative generic in =r ,n,m . We show that S (vi) is relative generic in =r ,n,m . Since S (v) is contained in S (vi) , Proposition 2.3(ii) yields that the condition = n + m or r = 0 is sufficient for S (vi) to be relative generic in =r ,n,m . It remains to show necessity of this condition. If = n + m and r = 0, then it is easy to see that S c (vi) ∩ =r ,n,m contains an inner point. Therefore S (vi) cannot be relative generic in =r ,n,m . This concludes the proof of the proposition.