# Universal regular control for generic semilinear systems

## Abstract

We consider discrete-time projective semilinear control systems \(\xi _{t+1} = A(u_t) \cdot \xi _t\), where the states \(\xi _t\) are in projective space \(\mathbb {R}\hbox {P}^{d-1}\), inputs \(u_t\) are in a manifold \(\mathcal {U}\) of arbitrary finite dimension, and \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\) is a differentiable mapping. An input sequence \((u_0,\ldots ,u_{N-1})\) is called universally regular if for any initial state \(\xi _0 \in \mathbb {R}\hbox {P}^{d-1}\), the derivative of the time-\(N\) state with respect to the inputs is onto. In this paper, we deal with the universal regularity of constant input sequences \((u_0, \ldots , u_0)\). Our main result states that generically in the space of such systems, for sufficiently large \(N\), all constant inputs of length \(N\) are universally regular, with the exception of a discrete set. More precisely, the conclusion holds for a \(C^2\)-open and \(C^\infty \)-dense set of maps \(A\), and \(N\) only depends on \(d\) and on the dimension of \(\mathcal {U}\). We also show that the inputs on that discrete set are nearly universally regular; indeed, there is a unique non-regular initial state, and its corank is 1. In order to establish the result, we study the spaces of bilinear control systems. We show that the codimension of the set of systems for which the zero input is not universally regular coincides with the dimension of the control space. The proof is based on careful matrix analysis and some elementary algebraic geometry. Then the main result follows by applying standard transversality theorems.

### Keywords

Discrete-time systems Semilinear systems Bilinear systems Universal regular control## 1 Introduction

### 1.1 Basic definitions and some questions

*accessible*from \(x_0\) in time \(N\) if the set \(\phi _N(\{x_0\} \,\times \, \mathcal {U}^N)\) of final states that can be reached from the initial state \(x_0\) has nonempty interior.

The implicit function theorem gives a sufficient condition for accessibility. If the derivative of the map \(\phi _N(x_0; \cdot )\) at input \((u_0,\ldots ,u_{N-1})\) is an onto linear map, then we say that the trajectory determined by \((x_0; u_0, \ldots , u_{N-1})\) is *regular*. So the existence of such a regular trajectory implies that the system is accessible from \(x_0\) in time \(N\).

Let us call an input \((u_0, \ldots , u_{N-1})\)*universally regular* if for every \(x_0 \in \mathcal {X}\), the trajectory determined by \((x_0; u_0, \ldots , u_{N-1})\) is regular; otherwise, the input is called *singular*.

The concept of universal regularity is central in this paper; it was introduced by Sontag in [13] in the context of continuous-time control systems. The discrete-time analog was considered by Sontag and Wirth in [14]. They showed that if the system (1.1) is accessible from every initial condition \(x_0\) in uniform time \(N\), then universally regular inputs do exist, provided one assumes the map \(F\) to be analytic. In fact, under those hypotheses, they showed that universally regular inputs are abundant: in the space of inputs of sufficiently large length, singular ones form a set of positive codimension.

*semilinear control system*.

*projective semilinear control system*. The projectivized system is also a useful tool for the study of the original system (1.3): see e.g., [5, 15].

Universally regular inputs for projective semilinear control systems were first considered by Wirth in [15]. Under his working hypotheses, the existence and abundance of such inputs is guaranteed by the aforementioned result of [14]; then he uses universally regular inputs to obtain global controllability properties.

The purpose of this paper is to establish results on the existence and abundance of universally regular inputs for projective semilinear control systems. Different from [14, 15], we will not necessarily assume our systems to be analytic. Let us consider systems (1.4) with \(\mathbb {K}=\mathbb {R}\) and \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\), a map of class \(C^r\), for some fixed \(r\geqslant 1\). To compensate for less rigidity, we do not try to obtain results that work for all \(C^r\) maps \(A\), but only for *generic* ones, i.e., those maps in a residual (dense \(G_\delta \)) subset, or, even better, in an open dense subset.

To make things more precise, assume \(\mathcal {U}\) is a \(C^\infty \) (real) manifold without boundary. All manifolds are assumed to be Hausdorff paracompact with a countable base of open sets, and of finite dimension. We will always consider the space \(C^r(\mathcal {U}, \hbox {GL}(d,\mathbb {R}))\) endowed with the strong \(C^r\) topology (which coincides with the usual uniform \(C^r\) topology in the case that \(\mathcal {U}\) is compact).

Hence, the first question we pose is this:

Taking \(N\) sufficiently large, is it true that for \(C^r\)-generic maps \(A\), the set of universally regular inputs in \(\mathcal {U}^N\) is itself generic?

It turns out that this question has a positive answer. Actually, in a work in preparation, we show that for generic maps \(A\), all inputs in \(\mathcal {U}^N\) are universally regular, except for those in a stratified closed set of positive codimension. So another natural question is this:

Fixed parameters \(d\), \(\dim \mathcal {U}\), \(N\), and \(r\), what is the minimum codimension of the set of singular inputs in \(\mathcal {U}^N\) that can occur for \(C^r\)-generic maps \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\)?

In full generality, this question seems to be very difficult. A simpler setting would be to restrict to *non-resonant inputs*, namely those inputs \((u_0,\ldots ,u_{N-1})\) such that \(u_i \ne u_j\) whenever \(i \ne j\). In this paper, we consider the most resonant case. Define a *constant* input of length \(N\) as an element of \(\mathcal {U}^N\) of the form \((u_0, u_0, \ldots , u_0)\). We propose ourselves to study universal regularity of inputs of this form.

### 1.2 The main result

We prove that generically the singular constant inputs form a very small set:

**Theorem 1.1**

Given \(d\geqslant 2\) and \(m \geqslant 1\), there exists an integer \(N\) with \(1 \leqslant N \leqslant d^2\) such that the following properties hold. Let \(\mathcal {U}\) be a smooth \(m\)-dimensional manifold without boundary. Then there exists a \(C^2\)-open \(C^\infty \)-dense subset \(\mathcal {O}\) of \(C^2(\mathcal {U}, \hbox {GL}(d,\mathbb {R}))\) such that for every system (1.4) with \(A \in \mathcal {O}\), all constant inputs of length \(N\) are universally regular, except for those in a zero-dimensional (i.e., discrete) set.

By saying that a subset \(\mathcal {O}\) of \(C^2(\mathcal {U}, \hbox {GL}(d,\mathbb {R}))\) is \(C^\infty \)-dense, we mean that for all \(r \geqslant 2\), the intersection of \(\mathcal {O}\) with \(C^r(\mathcal {U}, \hbox {GL}(d,\mathbb {R}))\) is dense in \(C^r(\mathcal {U}, \hbox {GL}(d,\mathbb {R}))\).

It is remarkable that the generic dimension of the set of singular constant inputs (namely, 0) does not depend on the dimension \(m\) of the control space \(\mathcal {U}\), neither on the dimension \(d-1\) of the state space. A partial explanation for this phenomenon is the following: First, the obstruction to universal regularity of the input \((u,u,\ldots ,u)\) is the combined degeneracy of the matrix \(A(u)\) and of the derivatives of \(A\) at \(u\). If \(m\) is small, then the image of the generic map \(A\) will avoid too degenerate matrices, which increases the chances of obtaining universal regularity. If \(m\) is large, then more degenerate matrices \(A(u)\) will inevitably appear; however, the large number of control parameters compensates, so universal control is still likely.

The singular inputs that appear in Theorem 1.1 are not only rare; we also show that they are “almost” universally regular:

**Theorem 1.2**

- 1.
There is a single direction \(\xi _0 \in \mathbb {R}\hbox {P}^{d-1}\) for which the corresponding trajectory of system (1.4) is not regular.

- 2.
The derivative of the map \(\phi _N(\xi _0; \cdot )\) at input \((u,\ldots ,u)\) has corank 1.

To sum up, for generic systems (1.4), the universal regularity of constant inputs can fail only in the weakest possible way: there is at most one non-regular state, which can be moved in all directions but one.

We actually describe precisely in Appendix E (ESM) the singular inputs that appear in Theorem 1.2. We show that these singular inputs can be unremovable by perturbations, and therefore, Theorem 1.1 is optimal in the sense that there are \(C^2\)-open (actually even \(C^1\)-open) sets of maps \(A\) for which the set of singular constant inputs is nonempty. Also, by \(C^1\)-perturbing any \(A\) in those \(C^2\)-open sets, one can obtain an infinite number of singular constant inputs. In particular, the set \(\mathcal {O}\) in the statement of the Theorem 1.1 is not \(C^1\)-open in general.

### 1.3 Reduction to the study of the set of poor data

The bulk of the proof of Theorem 1.1 consists on the computation of the dimension of certain canonical sets, as we now explain.

We fix \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {K})\) and consider the projective semilinear system (1.4). By the chain rule, the universal regularity of an input \((u_0, u_1, \ldots , u_{N-1})\) depends only on the 1-jets of \(A\) at points \(u_0\), ..., \(u_{N-1}\), i.e., on the first order Taylor approximations of \(A\) around those points.

*bilinear control system*(see [6]). For these systems, the zero input is a distinguished one and the focus of more attention.

*normalized derivatives*\(B_j = C_j A^{-1}\), which intrinsically take values in the Lie algebra \(\mathfrak {gl}(d,\mathbb {K})\). Consider the matrix datum \(\mathbf {A}= (A, B_{1}, \ldots , B_{m})\). We will explain how the universal regularity of the zero input is expressed in linear algebraic terms. Recall that the

*adjoint operator*of \(A\) acts on \(\mathfrak {gl}(d,\mathbb {K})\) by the formula \(\hbox {Ad}_A(B) = A B A^{-1}\). Consider the linear subspace \(\Lambda _N(\mathbf {A})\) of \(\mathfrak {gl}(d,\mathbb {K})\) spanned by the matrices

**Proposition 1.3**

The constant input \((0,\ldots ,0)\) of length \(N\) is universally regular for system (1.5) if and only if the space \(\Lambda _N(\mathbf {A})\) is transitive.

Here we say that a subspace of \(d \times d\) matrices with entries in the field \(\mathbb {K}\) is *transitive* if it acts transitively in the set \(\mathbb {K}^d_*\) of nonzero vectors.

Clearly, the spaces \(\Lambda _N(\mathbf {A})\) form a nested sequence that stabilizes to a space \(\Lambda (\mathbf {A})\) at some time \(N \leqslant d^2\). If \(\Lambda (\mathbf {A})\) is transitive, then the datum \(\mathbf {A}\) is called *rich*; otherwise it is called *poor*. Let \(\mathcal {P}_m^{(\mathbb {K})} = \mathcal {P}_{m,d}^{(\mathbb {K})}\) denote the set of poor data. A major part of our work is to study these sets. We prove:

**Theorem 1.4**

The set \(\mathcal {P}_{m}^{(\mathbb {R})}\) is closed and semialgebraic, and its codimension in \(\hbox {GL}(d,\mathbb {R}) \times (\mathfrak {gl}(d,\mathbb {R}))^m\) is \(m\).

**Theorem 1.5**

The set \(\mathcal {P}_{m}^{(\mathbb {C})}\) is algebraic, and its (complex) codimension in \(\hbox {GL}(d,\mathbb {C}) \times (\mathfrak {gl}(d,\mathbb {C}))^m\) is \(m\).

So Theorems 1.4 and 1.5 say how frequent universal regularity of the zero input is in the space of projective bilinear control systems (1.5).

### 1.4 Overview of the proofs

Theorem 1.1 follows rather directly from Theorem 1.4 by applying standard results from transversality theory. More precisely, the fact that the set \(\mathcal {P}_m^{(\mathbb {R})}\) is semialgebraic implies that it has a canonical stratification. This permits us to apply Thom’s jet transversality theorem and obtain Theorem 1.1.

On the other hand, Theorem 1.4 follows from its complex version Theorem 1.5 by simple abstract arguments.

Thus, everything is based on Theorem 1.5. One part of the result is easily obtained: we give examples of small disks of codimension \(m\) formed by poor data, so concluding that the codimension of \(\mathcal {P}_{m}^{(\mathbb {C})}\) is at most \(m\).

To prove the other inequality, one could try to exhibit an explicit codimension \(m\) set containing all poor data. For \(m=1\), this task is feasible (and we actually perform it, because with these conditions we can actually check universal regularity in concrete examples). However, for \(m=2\) already the task would be very laborious, and to expect to find a general solution seems unrealistic.

Our actual approach to prove the lower bound on the codimension of \(\mathcal {P}_{m}^{(\mathbb {C})}\) is indirect. Crudely speaking, after careful matrix computations, we find some sets in the complement of \(\mathcal {P}_{m}^{(\mathbb {C})}\) that are reasonably “large” (basically in terms of dimension). Then, by using some abstract results of algebraic geometry, we are able to show that \(\mathcal {P}_{m}^{(\mathbb {C})}\) is “small,” thus proving the other half of Theorem 1.5.

In fact, the results of this analysis are even better, and we conclude that the codimension inequality (1.6) is strict when \(k \geqslant 1\). This implies that poor data \((A, B_1, \ldots , B_m)\) for which the matrix \(A\) is degenerate form a subset of \(\mathcal {P}_{m}^{(\mathbb {C})}\) with strictly bigger codimension. Thus, we can show that the poor data that appear generically are well behaved, which leads to Theorem 1.2.

### 1.5 Holomorphic setting

In the case of complex matrices (i.e., \(\mathbb {K}= \mathbb {C}\)), we have a corresponding version of Theorem 1.1 where the maps \(A\) are holomorphic. Given an open subset \(\mathcal {U}\subset \mathbb {C}^m\), we denote by \(\mathcal {H}(\mathcal {U}, \hbox {GL}(d,\mathbb {C}))\) the set of holomorphic mappings \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {C})\) endowed with the usual topology of uniform convergence on compact sets.

**Theorem 1.6**

Given integers \(d\geqslant 2\) and \(m \geqslant 1\), there exists an integer \(N\geqslant 1\) with the following properties. Let \(\mathcal {U}\subset \mathbb {C}^m\) be open, and let \(K\subset \mathcal {U}\) be compact. Then there exists an open and dense subset \(\mathcal {O}\) of \(\mathcal {H}(\mathcal {U}, \hbox {GL}(d,\mathbb {C}))\) such that for any \(A \in \mathcal {O}\) the constant inputs in \(K^N\) are all universally regular for the system (1.4), except for a finite subset.

We have the straightforward corollary:

**Corollary 1.7**

Given integers \(d\geqslant 2\) and \(m \geqslant 1\), there exists an integer \(N\geqslant 1\) with the following properties. Let \(\mathcal {U}\subset \mathbb {C}^m\) be an open subset. There exists a residual subset \(\mathcal {R}\) of \(\mathcal {H}(\mathcal {U}, \hbox {GL}(d,\mathbb {C}))\) such that for any \(A \in \mathcal {R}\) the constant inputs in \(\mathcal {U}^N\) are all universally regular for the system (1.4), except for a discrete subset.

### 1.6 Directions for future research

One can also study uniform regularity of periodic inputs of higher period. Using our results for constant inputs, it is not difficult to derive some (non-sharp) codimension bounds for singular periodic inputs for generic systems. However, for highly resonant non-periodic inputs, we have no idea on how to obtain reasonable dimension estimates.

To obtain good estimates for the codimension of *non-resonant*, singular inputs for generic systems is relatively simpler from the point of view of matrix computations, but needs more sophisticated transversality theorems (e.g., multijet transversality). Since highly resonant inputs have large codimension themselves, it seems possible to obtain reasonably good codimension estimates for general inputs for generic systems.

Another interesting direction of research is to consider other Lie groups of matrices.

### 1.7 Organization of the paper

Section 2 contains some basic results about transitivity of spaces of matrices and its relation to universal regularity. We also obtain the easy parts of Theorems 1.4 and 1.5, namely (semi)algebraicity and the upper codimension inequalities.

In Sect. 3, we introduce the concept of rigidity, which is related to the quantity \(r(A)\) mentioned above. We state the central rigidity estimates (Theorem 3.6), which consist into two parts. The first and easier part is proved in the same Sect. 3, while the whole Sect. 4 is devoted to the proof of the second part.

Section 5 starts with some preliminaries in elementary algebraic geometry. Then we use the rigidity estimates to prove Theorem 1.5, following the strategy outlined above (§ 1.4). Theorem 1.4 follows easily. We also obtain a lemma that is needed for the proof of Theorem 1.2.

In Sect. 6, we deduce Theorem 1.1 from previous results and standard theorems stratifications and transversality.

The paper also has some appendices (Electronic Supplementary Material):

Appendix A (ESM) basically reobtains the major results in the special case \(m=1\), where we actually gain additional information of practical value: as mentioned in Sect. 1.4, it is possible to describe explicitly what 1-jets the map \(A\) should avoid in order to satisfy the conclusions of Theorems 1.1 and 1.2. The arguments necessary for the \(m=1\) case are much simpler and more elementary than those in Sects. 3, 4 and 5. Therefore, that appendix is also useful to give the reader some intuition about the general problem, and as a source of examples. Appendix A (ESM) is written in a slightly informal way, and it can be read after Sect. 2 (though the final part requires Lemma 3.1).

Appendix B (ESM) contains the proofs of necessary algebraic–geometric results, especially the one that allows us to obtain estimate (1.8).

Appendix C (ESM) reviews the necessary concepts and results on stratifications and proves a prerequisite transversality proposition.

In Appendix D (ESM), we apply Theorem 1.5 to prove a version of Theorem 1.1 for holomorphic mappings.

In Appendix E (ESM), we study the singular constant inputs of generic type, proving Theorem 1.2 and the other assertions made at the end of Sect.1.2 concerning the sharpness of Theorem 1.1. We also discuss the generic validity of some control-theoretic properties related to accessibility and regularity.

## 2 Preliminary facts on the poor data

In this section, we review some basic properties related to poorness and prove the easy inequalities in Theorems 1.4 and 1.5.

### 2.1 Transitive spaces

Let \(E\) and \(F\) be finite-dimensional vector spaces over the field \(\mathbb {K}\). Let \(\mathcal {L}(E,F)\) be the space of linear maps from \(E\) to \(F\). A vector subspace \(\Lambda \) of \(\mathcal {L}(E,F)\) is called *transitive* if for every \(v \in E {\backslash } \{0\}\), we have \(\Lambda \cdot v = F\), where \(\Lambda \cdot v = \{ L(v) ; \; L \in \Lambda \}\).

Under the identification \(\mathcal {L}(\mathbb {K}^n, \mathbb {K}^m) = \hbox {Mat}_{m \times n}(\mathbb {K})\), we may also speak of transitive spaces of matrices.

The following examples illustrate the concept; they will also be needed in later considerations.

*Example 2.1*

*Toeplitz matrix*, resp. a

*Hankel matrix*, is a matrix of the formThe set of Toeplitz matrices and the set of complex Hankel matrices constitute examples transitive subspaces of \(\mathfrak {gl}(d,\mathbb {K})\). Transitivity of the Toeplitz space is a particular case of Example 2.2, and transitivity of Hankel space follows from Remark 2.3. For \(\mathbb {K}= \mathbb {C}\), these spaces are optimal, in the sense that they have the least possible dimension; see [1].

*Example 2.2*

A *generalized Toeplitz space* is a subspace \(\Lambda \) of \(\hbox {Mat}_{d\times d}(\mathbb {K})\) (where \(d \geqslant 2\)) with the following property: For any two matrix entries \((i_1,j_1)\) and \((i_2,j_2)\) which are not in the same diagonal (i.e., \(i_1-j_1 \ne i_2-j_2\)), the linear map \((b_{i,j})_{i,j} \in \Lambda \mapsto (b_{i_1,j_1},b_{i_2,j_2}) \in \mathbb {C}^2\) is onto. Equivalently, a space is generalized Toeplitz if it can be defined by a number of linear relations between the matrix coefficients so that each relation involves only the entries on a same diagonal, and so that the relations do not force any matrix entry to be zero. We will prove later (see Sect. 3.3) that *every generalized Toeplitz space is transitive*.

*Remark 2.3*

If \(\Lambda \) is a transitive subspace of \(\mathcal {L}(E,F)\) and \(P \in \mathcal {L}(E,E)\), \(Q \in \mathcal {L}(F,F)\) are invertible operators, then \(P \cdot \Lambda \cdot Q := \{PLQ ; \; L \in \Lambda \}\) is a transitive subspace of \(\mathcal {L}(E,F)\).

A subset of \(\mathbb {K}^n\) is called

*algebraic*if it is expressed by polynomial equations with coefficients in \(\mathbb {K}\).A subset of \(\mathbb {R}^n\) is called

*semialgebraic*if it is the union of finitely many sets, each of them defined by finitely many real polynomial equations and inequalities (see [2, 3]).

**Proposition 2.4**

- 1.
The set \(\mathcal {N}_{m,n,k}^{(\mathbb {R})}\) is semialgebraic.

- 2.
The set \(\mathcal {N}_{m,n,k}^{(\mathbb {C})}\) is algebraic.

*Proof*

To see part 2, we take \(\mathbb {K}=\mathbb {C}\) and projectivize the \(\mathbb {C}^n_*\) fiber, obtaining an algebraic subset \([\hbox {Mat}_{m \times n}(\mathbb {C})]^k \times \mathbb {C}\hbox {P}^{n-1}\) whose projection along the \(\mathbb {C}\hbox {P}^{n-1}\) fiber is \(\mathcal {N}_{m,n,k}^{(\mathbb {C})}\). So part 2 follows from the fact that projections along projective fibers take algebraic sets to algebraic sets (see [12, p. 58]).

Complex transitivity of real matrices is a stronger property than real transitivity:

**Proposition 2.5**

The real part of \(\mathcal {N}_{m,n,k}^{(\mathbb {C})}\) (that is, its intersection with \([\hbox {Mat}_{m \times n}(\mathbb {R})]^k\)) contains \(\mathcal {N}_{m,n,k}^{(\mathbb {R})}\).

The proof is an easy exercise.

### 2.2 Universal regularity for constant inputs and richness

In this subsection, we prove Proposition 1.3; in fact we prove a more precise result, and also fix some notation.

If \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {C})\) is a differentiable map, then the *normalized derivative* of \(A\) at a point \(u\) is the linear map \(T_u \mathcal {U}\rightarrow \mathfrak {gl}(d,\mathbb {R})\) given by \(h \mapsto (DA(u)\cdot h)\circ A^{-1}(u)\).

Let \(\phi _N(\xi _0,\hat{u})\) be the state \(\xi _N\in \mathbb {K}\hbox {P}^d\) of the system (1.4) determined by the initial state \(\xi _0\) and the input sequence \(\hat{u}\in \mathcal {U}^N\). Let \(\partial _2 \phi _N(\xi _0,\hat{u})\) be the derivative of the map \(\phi _N(\xi _0, \cdot )\) at \(\hat{u}\).

**Proposition 2.6**

In particular (since \(A =A(u)\) is invertible), the input \(\hat{u}\) is universally regular if and only if \(\Lambda _N(\mathbf {A})\) is a transitive space, which is the statement of Proposition 1.3.

*Proof of Proposition 2.6*

### 2.3 The sets of poor data

For emphasis, we repeat the definition already given in the introduction: The datum \(\mathbf {A}= (A, B_1, \ldots , B_m) \in \hbox {GL}(d,\mathbb {K}) \times [\mathfrak {gl}(d,\mathbb {K})]^m\) is *rich* if the space \(\Lambda (\mathbf {A}) = \Lambda _{d^2}(\mathbf {A})\) is transitive, and *poor* otherwise. The concept in fact depends on the field under consideration. The set of such poor data is denoted by \(\mathcal {P}_{m,d}^{(\mathbb {K})}\).

It follows immediately from Proposition 2.4 that \(\mathcal {P}_{m,d}^{(\mathbb {R})}\) is a closed and semialgebraic subset of \(\hbox {GL}(d,\mathbb {R}) \times [\mathfrak {gl}(d,\mathbb {R})]^m\) and \(\mathcal {P}_{m,d}^{(\mathbb {C})}\) is an algebraic subset of \(\hbox {GL}(d,\mathbb {C}) \times [\mathfrak {gl}(d,\mathbb {C})]^m\). This proves part of Theorems 1.4 and 1.5.

*saturated*if \((A, B_1, \ldots , B_m) \in \mathcal {Z}\) implies that: \((A, B_1, \ldots , B_m) \in \mathcal {Z}\) implies that:

for all \(P \in \hbox {GL}(d,\mathbb {K})\) we have \((P^{-1}AP, P^{-1}B_1 P, \ldots , P^{-1}B_m P) \in \mathcal {Z}\);

for all \(Q \!=\! (q_{ij}) \!\in \! \hbox {GL}(m,\mathbb {K})\), letting \(B'_i \!=\! \sum \nolimits _j q_{ij} B_j\), we have \((A, B_1', \ldots , B_m')\!\in \!\mathcal {Z}\).

### 2.4 The easy codimension inequalities of Theorems 1.4 and 1.5

Here we will discuss the simplest examples of poor data.

*conspicuously poor*if there exists a change of bases \(P \in \hbox {GL}(d,\mathbb {K})\) such that:

the matrix \(P^{-1} A P\) is diagonal;

the matrices \(P^{-1} B_k P\) have a zero entry in a common off-diagonal position; more precisely, there are indices \(i_0\), \(j_0 \in \{1,\ldots ,d\}\) with \(i_0 \ne j_0\) such that for each \(k \in \{1,\ldots ,m\}\), the \((i_0,j_0)\) entry of the matrix \(P^{-1} B_k P\) vanishes.

**Lemma 2.7**

Conspicuously poor data are poor.

*Proof*

Let \(\mathbf {A}= (A, B_1, \ldots , B_m)\) be conspicuously poor. With a change of basis, we can assume that \(A\) is diagonal. Let \((e_1,\ldots ,e_d)\) be the canonical basis of \(\mathbb {K}^d\). Let \((i,j)\) be the entry position where all \(B_i\)’s have a zero entry. By (2.4), all matrices in the space \(\Lambda (\mathbf {A}) = \mathfrak {R}_{\hbox {Ad}_A}(\hbox {Id}, B_1, \ldots , B_m)\) have a zero entry in the \((i_0,j_0)\) position. In particular, there is no \(L \in \Lambda (\mathbf {A})\) such that \(L \cdot e_{j_0} = e_{i_0}\), showing that this space is not transitive.

The converse of this lemma is certainly false. (Many examples appear in Appendix A (ESM); see also Example 3.5.) However, we will see in § A.1 that the converse holds for generic \(A\).

We will use Lemma 2.7 to prove the easy codimension inequalities for Theorems 1.4 and 1.5; first we need to recall the following:

**Proposition 2.8**

Suppose \(A \in \hbox {Mat}_{d\times d}(\mathbb {K})\) is diagonalizable over \(\mathbb {K}\) and with simple eigenvalues only. Then there is a neighborhood of \(A\) where the eigenvalues vary smoothly, and where the eigenvectors can be chosen to vary smoothly.

**Proposition 2.9**

(Easy half of Theorems 1.4 and 1.5) For both \(\mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\), we have \({{\mathrm{codim }}}_{\mathbb {K}} \mathcal {P}^{(\mathbb {K})}_m \leqslant m\).

*Proof*

Using Proposition 2.8, we can exhibit smoothly embedded disks of codimension \(m\) inside \(\hbox {GL}(d,\mathbb {K}) \times \mathfrak {gl}(d,\mathbb {K})^m\) formed by conspicuously poor data.\(\square \)

## 3 Rigidity

The aim of this section is to state Theorem 3.6 and prove its first part. Along the way, we will establish several lemmas which will be reused in the proof of the second part of the theorem in Sect. 4.

### 3.1 Acyclicity

Consider a linear operator \(H :E \rightarrow E\), where \(E\) is a finite-dimensional complex vector space. The *acyclicity* of \(H\) is defined as the least number \(n\) of vectors \(v_1\), ..., \(v_n \in E\) such that \(\mathfrak {R}_H(v_1, \ldots , v_n) = E\). We denote \(n = {{\mathrm{acyc }}}H\). If \(n = 1\), then \(H\) is called a *cyclic operator*, and \(v_1\) is called a *cyclic vector*.

**Lemma 3.1**

*Proof*

View \(E\) as a module over the ring of polynomials \(\mathbb {C}[x]\) by defining \(xv=H(v)\) for \(v \in E\). Then the lemma follows from [11, Theorem 6.4].

The *geometric multiplicity* of an eigenvalue \(\lambda \) of \(H\) is the dimension of the kernel of \(H - \lambda \hbox {Id}\) (or, equivalently, the number of corresponding Jordan blocks).

**Proposition 3.2**

The acyclicity of an operator equals the maximum of the geometric multiplicities of its eigenvalues.

*Proof*

This follows from the Primary Cyclic Decomposition Theorem together with Lemma 3.1. \(\square \)

*Remark 3.3*

The operators which interest us most are \(H = \hbox {Ad}_A\), where \(A \in \hbox {GL}(d,\mathbb {C})\). It is useful to observe that *the geometric multiplicity of 1 as an eigenvalue of*\(\hbox {Ad}_A\)*equals the codimension of the conjugacy class of*\(A\)*inside*\(\hbox {GL}(d,\mathbb {C})\). To prove this, consider the map \(\Psi _A :\hbox {GL}(d,\mathbb {C}) \rightarrow \hbox {GL}(d,\mathbb {C})\) given by \(\Psi _A(X) = \hbox {Ad}_X(A)\). The derivative at \(X=\hbox {Id}\) is \(H \mapsto HA-AH\); so \({{\mathrm{Ker }}}D\Psi _A (\hbox {Id}) = {{\mathrm{Ker }}}(\hbox {Ad}_A - \hbox {id})\). Therefore, when \(X=\hbox {Id}\), the rank of \(D\Psi _A (X)\) equals the geometric multiplicity of 1 as an eigenvalue of \(\hbox {Ad}_A\). To see that this is true for any \(X\), notice that \(\Psi _A = \Psi _{\hbox {Ad}_X(A)} \circ R_{X^{-1}}\) (where \(R\) denotes a right-multiplication diffeomorphism of \(\hbox {GL}(d,\mathbb {C})\)).

We will see later (Lemma 4.11) that 1 is the eigenvalue of \(\hbox {Ad}_A\) with the biggest geometric multiplicity. By Proposition 3.2, we conclude that \({{\mathrm{acyc }}}\hbox {Ad}_A\) equals the codimension of the conjugacy class of \(A\).

### 3.2 Definition of rigidity, and the main rigidity estimate

*rigidity*of \(H\), denoted \({{\mathrm{rig }}}H\), as the least \(n\) such that there exist \(L_1\), ..., \(L_n \in \mathcal {L}(E,F)\) so that \(\mathfrak {R}_H(L_1 , \ldots , L_n)\) is transitive. Therefore,

*modified rigidity*of \(H\), denoted \({{\mathrm{rig }}}_+ H\). The definition is the same, with the difference that if \(E = F\), then \(L_1\) is required to be the identity map in \(\mathcal {L}(E,E)\). Of course,

*Example 3.4*

If \(A\in \hbox {GL}(d,\mathbb {C})\) is unconstrained (see § A.1), then \({{\mathrm{rig }}}_+ \hbox {Ad}_A = 2\). Indeed, if we take a matrix \(B \in \mathfrak {gl}(d,\mathbb {C})\) whose expression in the base that diagonalizes \(A\) has no zeros off the diagonal, then, by Lemma A.1, \(\Lambda (A,B) = \mathfrak {R}_{\hbox {Ad}_A}(\hbox {Id},B)\) is rich.

More generally, if \(A\in \hbox {GL}(d,\mathbb {C})\) is little constrained (see Appendix A in ESM), then it follows from Proposition A.3 that \({{\mathrm{rig }}}_+ \hbox {Ad}_A = 2\).

*Example 3.5*

Consider \(A = \hbox {Diag}(1, \alpha , \alpha ^2)\) where \(\alpha = e^{2\pi i /3}\). (In the terminology of § A.1, \(A\) has constraints of type 1.) Since \(\hbox {Ad}_A^3\) is the identity, we have \(\dim \mathfrak {R}_{\hbox {Ad}_A}(\hbox {Id},B) \leqslant 4\) for any \(B \in \mathfrak {gl}(3,\mathbb {C})\). By the result of Azoff [1] already mentioned at Example 2.1, the minimum dimension of a transitive subspace of \(\mathfrak {gl}(3,\mathbb {C})\) is 5. This shows that \({{\mathrm{rig }}}_+ \hbox {Ad}_A \geqslant 3\). (Actually, equality holds, as we will see in Example 3.9 below.)

*equivalent mod*\(T\).

**Theorem 3.6**

- 1.
If \(c(A) = d\), then \({{\mathrm{rig }}}_+ \hbox {Ad}_A = 2\).

- 2.
If \(c(A) < d\), then \({{\mathrm{rig }}}_+ \hbox {Ad}_A \leqslant {{\mathrm{acyc }}}\hbox {Ad}_A - c(A) + 1\).

*Remark 3.7*

When \(c(A) = d\), we have \({{\mathrm{acyc }}}\hbox {Ad}_A = d\) (this will follow from Lemma 4.11); so the conclusion of part 2 does not hold in this case.

*Remark 3.8*

The conditions of \(A\) being unconstrained and \(A\) having \(c(A)=d\) both mean that \(A\) is “non-degenerate.” Both of them imply small rigidity, according to Example 3.4 and part 1 of Theorem 3.6. It is important, however, not to confuse the two properties; in fact, none implies the other.

*Example 3.9*

Consider again \(A\) as in Example 3.5. The eigenvalues of \(\hbox {Ad}_A\) are 1, \(\alpha \), and \(\alpha ^2\), each with multiplicity 3; so Proposition 3.2 gives \({{\mathrm{acyc }}}\hbox {Ad}_A = 3\). So Theorem 3.6 tell us that \({{\mathrm{rig }}}_+ \hbox {Ad}_A \leqslant 3\), which is actually sharp.

The proof of part 1 of Theorem 3.6 will be given in § 3.5 after a few preliminaries (Sect. 3.3 and 3.4). These preliminaries are also used in the proof of the harder part 2, which will be given in Sect. 4.

### 3.3 A criterion for transitivity

We will show the transitivity of certain spaces of matrices that remotely resemble Toeplitz matrices.

**Lemma 3.10**

Assume that \(\Lambda ^{[\mathtt {R}]}\) is transitive for each \(\mathtt {R}\in \mathcal {R}\). Then \(\Lambda \) is transitive.

An interesting feature of the lemma which will be useful later is that it can be applied recursively. Before giving the proof of the lemma, we illustrate its usefulness by showing the transitivity of generalized Toeplitz spaces:

*Proof of Example 2.2*

Consider the partition of \([1,d]^2\) into \(1 \times 1\) “rectangles.” If \(\Lambda \) is a generalized Toeplitz space then \(\Lambda ^{[\mathtt {R}]} = \hbox {Mat}_{1\times 1}(\mathbb {C}) = \mathbb {C}\) for each rectangle \(\mathtt {R}\). These are transitive spaces, so Lemma 3.10 implies that \(\Lambda \) is transitive. \(\square \)

Before proving Lemma 3.10, notice the following dual characterization of transitivity, whose proof is immediate:

**Lemma 3.11**

A subspace \(\Lambda \subset \mathcal {L}(\mathbb {C}^s,\mathbb {C}^t)\) is transitive iff for any nonzero vector \(u \in \mathbb {C}^s\) and any nonzero linear functional \(\phi \in (\mathbb {C}^t)^*\) there exists \(M \in \Lambda \) such that \(\phi (M \cdot u) \ne 0\).

*Proof of Lemma 3.10*

Let \(j_0\) be the least index such that \(u_{j_0} \ne 0\), and let \(i_0\) be the greatest index such that \(\phi _{i_0} \ne 0\). Let \(\mathtt {R}\) be the element of \(\mathcal {R}\) that contains \((i_0,j_0)\). Notice that if \(M\) is of the form (3.3), then the \((i,j)\)-entries of \(M\) that are above left (resp. below right) of \(\mathtt {R}\) do not contribute to the sum (3.5), because \(u_j\) (resp. \(\phi _i\)) vanishes. That is, \(\phi (M \cdot u)\) depends only on \(M_{\mathtt {R}}\) and is given by \(\sum _{(i,j)\in \mathtt {R}} \phi _i x_{ij} u_j\); Since \(\Lambda ^{[\mathtt {R}]}\) is transitive, by Lemma 3.11 there is a choice of a matrix \(M \in \Lambda \) of the form (3.3) so that \(\phi (M \cdot u) \ne 0\). So we are done.

### 3.4 Preorder in the complex plane

We consider the set \(\mathbb {C}_* / T\) of equivalence classes of the relation (3.1). Since \(T\) is the torsion subgroup of \(\mathbb {C}_*\), the quotient \(\mathbb {C}_* / T\) is an abelian torsion-free group.

**Proposition 3.12**

There exists a multiplication-invariant total order \(\preccurlyeq \) on \(\mathbb {C}_* / T\).

The proposition follows from a result of Levi [10], but nevertheless let us give a direct proof:

*Proof*

There is an isomorphism between \(\mathbb {R}\oplus (\mathbb {R}/ \mathbb {Q})\) and \(\mathbb {C}_*/T\), namely \((x,y) \mapsto \exp (x + 2\pi i y)\). So it suffices to find a multiplication-invariant order in \(\mathbb {R}/\mathbb {Q}\) (and then take the lexicographic order). Take a Hamel basis \(B\) of the \(\mathbb {Q}\)-vector space \(\mathbb {R}\) so that \(1 \in B\). Then \(\mathbb {R}/\mathbb {Q}\) is a direct sum of abelian groups \(\bigoplus _{x\in B, x \ne 1} x\mathbb {Q}\). Order each \(x\mathbb {Q}\) in the usual way and take any total order on \(B\). Then the induced lexicographic order on \(\mathbb {R}/\mathbb {Q}\) is multiplication-invariant, and the proof is concluded.\(\square \)

\(z \preccurlyeq z'\) or \(z' \preccurlyeq z\);

\(z \preccurlyeq z'\) and \(z' \preccurlyeq z\)\(\Longleftrightarrow \)\(z \asymp z'\);

\(z \preccurlyeq z'\) and \(z' \preccurlyeq z''\)\(\Longrightarrow \)\(z \preccurlyeq z''\);

\(z \preccurlyeq z'\)\(\Longrightarrow \)\(z z'' \preccurlyeq z' z''\).

\(z \preccurlyeq z'\)\(\Longrightarrow \)\((z')^{-1} \preccurlyeq z^{-1}\).

### 3.5 Proof of the easy part of Theorem 3.6

*Proof of part 1 of Theorem 3.6*

## 4 Proof of the hard part of the rigidity estimate

This section is wholly devoted to proving part 2 of Theorem 3.6. In the course of the proof, we need to introduce some terminology and to establish several intermediate results. None of these are used in the rest of the paper, apart form a simple consequence, which is Remark 4.12.

### 4.1 The normal form

Let \(A \in \hbox {GL}(d,\mathbb {C})\). In order to describe the estimate on \({{\mathrm{rig }}}_+ \hbox {Ad}_A\), we need to put \(A\) in a certain normal form, which we now explain. Fix a preorder \(\preccurlyeq \) on \(\mathbb {C}_*\) as in § 3.4.

### 4.2 Rectangular partitions

This subsection contains several definitions that will be fundamental in all arguments until the end of the section. We will define certain subregions of the set \(\{1,\dots ,d\}^2\) of matrix entry positions that depend on the normal form of the matrix \(A\). Later we will see they are related to \(\hbox {Ad}_A\)-invariant subspaces. Those regions will be c-rectangles, e-rectangles, and j-rectangles (where c stands for classes of eigenvalues, e for eigenvalues and j for Jordan blocks). Regions will have some numerical attributes (banners and weights) coming from their geometry and from the eigenvalues of \(A\) they will be associated with. Those attributes will be related to numerical invariants of \(\hbox {Ad}_A\) (eigenvalues and geometric multiplicities), but we use different names so that we remember their geometric meaning and so that they are not mistaken for the corresponding invariants of \(A\). We also introduce positional attributes of the regions (arguments and latitudes) which will be useful fundamental later in the proofs of our rigidity estimates.

The partition \(\mathcal {P}_\mathrm{c}\) corresponds to equivalence classes of eigenvalues under the relation \(\asymp \), that is, the right endpoints of its atoms are the numbers \(s_1 + \cdots + s_k\) where \(k=r\) or \(k\) is such that \(\lambda _k \prec \lambda _{k+1}\).

The partition \(\mathcal {P}_\mathrm{e}\) corresponds to eigenvalues: the right endpoints of its atoms are the numbers \(s_1 + \cdots + s_k\), where \(1 \leqslant k \leqslant r\). So \(\mathcal {P}_\mathrm{e}\) refines \(\mathcal {P}_\mathrm{c}\).

The partition \(\mathcal {P}_\mathrm{j}\) corresponds to Jordan blocks: the right endpoints of its atoms are the numbers \(s_1 + \cdots + s_{k-1} + t_{k,1} + \cdots + t_{k,\ell }\), where \(1 \leqslant k \leqslant r\) and \(1 \leqslant \ell \leqslant \tau _k\). So \(\mathcal {P}_\mathrm{j}\) refines \(\mathcal {P}_\mathrm{e}\).

*c-rectangles*, the elements of \(\mathcal {P}_\mathrm{e}^2\) are called

*e-rectangles*, and elements of \(\mathcal {P}_\mathrm{j}^2\) are called

*j-rectangles*. Thus, the square \([1,d]^2\) is a disjoint union c-rectangles, each of them is a disjoint union of e-rectangles, each of them is a disjoint union of j-rectangles.

*Example 4.1*

For each e-rectangle, we define its *row eigenvalue* and its *column eigenvalue* in the obvious way: If an e-rectangle \(\mathtt {E}\) equals \(I_k \times I_\ell \) where \(I_k\) and \(I_\ell \) are intervals with right endpoints \(s_1 + \cdots + s_k\) and \(s_1 + \cdots + s_\ell \), respectively, then the row eigenvalue of \(\mathtt {E}\) is \(\lambda _k\) and the column eigenvalue of \(\mathtt {E}\) is \(\lambda _\ell \). The row and column eigenvalues of a j-rectangle \(\mathtt {J}\) are defined, respectively, as the row and column eigenvalues of the e-rectangle that contains it.

Let \(\mathtt {E}\) be an e-rectangle with row eigenvalue \(\lambda _k\) and column eigenvalue \(\lambda _\ell \). The *banner* of \(\mathtt {E}\) is defined by \(\lambda _k^{-1}\lambda _\ell \). The *argument* of the e-rectangle is the quantity \(\theta _\ell - \theta _k \in (-2\pi ,2\pi )\). It coincides modulo \(2\pi \) with the argument of the banner, but it contains more information than the argument of the banner.

Each j-rectangle \(\mathtt {J}\) has an address of the type “\(i\)th row, \(j\)th column, e-rectangle \(\mathtt {E}\)”; then the *latitude* of the j-rectangle \(\mathtt {J}\) within the e-rectangle \(\mathtt {E}\) is defined as \(j-i\). See an example in Fig. 1.

If two e-rectangles lie in the same c-rectangle, then their banners are equivalent mod \(T\). Thus, every c-rectangle has a well-defined *banner class* in \(\mathbb {C}^*/T\).

If a j-rectangle, e-rectangle, or c-rectangle intersects the diagonal \(\{(1,1), \ldots ,\)\( (d,d)\}\), then we call it *equatorial*. Equatorial regions are always square. Thus, every equatorial e-rectangle has banner 1.

The *weight* of a j-rectangle is defined as the minimum of its sides. The weight of a union \(R\) of j-rectangles in \([1,d]^2\) is defined as the sum of the weights of those j-rectangles. We denote it by \({{\mathrm{wgt }}}R\). We can in particular consider the weights of e and c-rectangles, and of the complete square \([1,d]^2\).

Let us notice some facts on the location of the banners (which will be useful to apply Lemma 3.10):

**Lemma 4.2**

- 1.
All the c-rectangles with banner class \([\beta ]\) are inside the rectangles marked with \(\times \).

- 2.
If the e-rectangle \(\mathtt {E}\) has nonnegative (resp. negative) argument, then the all the e-rectangles with nonnegative (resp. negative) argument and with same banner \(\beta \) are inside the rectangles marked with \(*\).

*Proof*

In view of the ordering of the eigenvalues (4.1), the banner class increases strictly (with respect to the order \(\prec \), of course) when we move rightwards or upwards to another c-rectangle. So Claim (1) follows.

The argument of an e-rectangle takes values in the interval \((-2\pi ,2\pi )\). It increases strictly by moving rightwards or upwards inside \(\mathtt {C}\). If two e-rectangles in the same c-rectangle have both nonnegative or negative argument, then they have the same banner if and only if they have the same argument. So Claim (2) follows.

### 4.3 The action of the adjoint of \(A\)

*submatrix*of \(X\) corresponding to \(\mathtt {R}\) as \((x_{i,j})_{(i,j) \in \mathtt {R}}\). We regard the space of \(\mathtt {R}\)-submatrices as \(\mathcal {L} \big ( \{0\}^{q-1} \times \mathbb {C}^s \times \{0\}^{d-q-s+1} , \{0\}^{p-1} \times \mathbb {C}^t \times \{0\}^{d-p-t+1} \big )\), or as the set of \(d \times d\) matrices whose entries outside \(\mathtt {R}\) are all zero. Such spaces are denoted by \(\mathtt {R}^\square \), and are invariant under \(\hbox {Ad}_A\). Indeed, if \(\mathtt {R}= \mathtt {J}\) is a j-rectangle, then identifying \(\mathtt {J}^\square \) with \(\hbox {Mat}_{t\times s}(\mathbb {C})\), the action of \(\hbox {Ad}_A | \mathtt {J}^\square \) is given by

**Lemma 4.3**

For each j-rectangle \(\mathtt {J}\), the only eigenvalue of \(\hbox {Ad}_A | \mathtt {J}^\square \) is the banner of the e-rectangle that contains \(\mathtt {J}\). Moreover, the geometric multiplicity of the eigenvalue is the weight of the j-rectangle.

*Proof*

The matrix of the linear operator \(\hbox {Ad}_A | \mathtt {J}^\square \) can be described using the Kronecker product: see [9, Lemma 4.3.1]. The Jordan form of this operator is then described by [9, Theorem 4.3.17(a)]. The assertions of the lemma follow.

The eigenvalues of \(\hbox {Ad}_A\) are the banners of e-rectangles.

The geometric multiplicity of the eigenvalue \(\beta \) for \(\hbox {Ad}_A\) is the total weight of e-rectangles of banner \(\beta \).

*identity on*\(\mathtt {R}^\square \). The following observation will be useful:

**Lemma 4.4**

If \(\mathtt {J}\) is an equatorial j-rectangle, then the identity on \(\mathtt {J}^\square \) is an eigenvector of the operator \(\hbox {Ad}_A | \mathtt {J}^\square \) corresponding to a Jordan block of size \(1 \times 1\).

*Proof*

Suppose \(\mathtt {J}\) has size \(t \times t\) and row (or column) eigenvalue \(\lambda \). Assume that the claim is false. This means that there exists a matrix \(X \in \hbox {Mat}_{t \times t}(\mathbb {C})\) such that \(J_t(\lambda ) X J_t(\lambda )^{-1} = X + \hbox {Id}\), which is impossible because \(X\) and \(X + \hbox {Id}\) have different spectra.

### 4.4 Rigidity estimates for j-rectangles and e-rectangles

**Lemma 4.5**

For any j-rectangle \(\mathtt {J}\), we have \({{\mathrm{rig }}}_+ (\hbox {Ad}_A | \mathtt {J}^\square ) \leqslant {{\mathrm{wgt }}}\mathtt {J}\).

*Proof*

By Lemma 4.3 (and Proposition 3.2), \(\hbox {Ad}_A|\mathtt {J}^\square \) has acyclicity \(n = {{\mathrm{wgt }}}\mathtt {J}\), that is, there are matrices \(X_1\), ..., \(X_n \in \mathtt {J}^\square \) such that \(\mathfrak {R}_{\hbox {Ad}_A}(X_1, \ldots , X_n)\) is the whole \(\mathtt {J}^\square \) (and, in particular, is transitive in \(\mathtt {J}^\square \)). So \({{\mathrm{rig }}}(\hbox {Ad}_A|\mathtt {J}^\square ) \leqslant n\), which proves the lemma for non-equatorial j-rectangles.

If \(\mathtt {J}\) is an equatorial j-rectangle then, by Lemma 4.4, \(\mathtt {J}^\square \) splits invariantly into two subspaces, one of them spanned by the identity matrix on \(\mathtt {J}^\square \). So we can choose the matrices \(X_i\) above so that \(X_1\) is the identity. This shows that \({{\mathrm{rig }}}_+ (\hbox {Ad}_A|\mathtt {J}^\square ) \leqslant n\).

In all that follows, we adopt the convention \(\max \varnothing = 0\).

**Lemma 4.6**

*Proof*

the submatrix \(N_\mathtt {J}\) equals \(M\);

for every j-rectangle \(\mathtt {J}'\) in \(\mathtt {E}\) that has a different latitude than \(\mathtt {J}\), the submatrix \(N_{\mathtt {J}'}\) vanishes.

In notation (3.4), the claim we have just proved means that \(\Delta ^{[\mathtt {J}]} \supset \Lambda _\mathtt {J}\). So we can apply Lemma 3.10 and conclude that \(\Delta \) is a transitive subspace of \(\mathtt {E}^\square \). Therefore, \({{\mathrm{rig }}}_+ (\hbox {Ad}_A | \mathtt {E}^\square ) \leqslant \sum n_\ell \), as we wanted to show.

*Example 4.7*

Using Lemmas 4.5 and 4.6, we see that the e-rectangle \(\mathtt {E}\) whose j-rectangle weights are indicated in Fig. 1 has \({{\mathrm{rig }}}_+ (\hbox {Ad}_A | \mathtt {E}^\square ) \leqslant 5\).

In fact, we will not use Lemmas 4.5 and 4.6 directly, but only the following immediate consequence:

**Lemma 4.8**

For every e-rectangle \(\mathtt {E}\) we have \({{\mathrm{rig }}}_+ (\hbox {Ad}_A | \mathtt {E}^\square ) \leqslant {{\mathrm{wgt }}}\mathtt {E}\). The inequality is strict if \(\mathtt {E}\) has more than one row of j-rectangles and more that one column of j-rectangles.

### 4.5 Comparison of weights

If \(\mathtt {R}\) is a j-rectangle, e-rectangle or c-rectangle, we define its *row projection*\(\pi _{\mathrm{r}}(\mathtt {R})\) as the unique equatorial j-rectangle, e-rectangle or c-rectangle (respectively) that is in the same row as \(\mathtt {R}\). Analogously, we define the *column projection*\(\pi _{\mathrm{c}}(\mathtt {R})\).

**Lemma 4.9**

This is a clear consequence of the abstract lemma below, taking \(x_\alpha \), \(\alpha \in F_0\) (resp. \(\alpha \in F_1\)) as the sequence of heights (resp. widths) of j-rectangles in \(\mathtt {E}\), counting repetitions.

**Lemma 4.10**

*Proof*

If \(\mathtt {R}\) is a c-rectangle or the entire square \([1,d]^2\), let \({{\mathrm{wgt }}}_1 \mathtt {R}\) denote the sum of the weights of the e-rectangles in \(\mathtt {R}\) with banner 1.

Let us give the following useful consequence of Lemma 4.9:

**Lemma 4.11**

\({{\mathrm{acyc }}}\hbox {Ad}_A = {{\mathrm{wgt }}}_1 [1,d]^2\).

*Proof*

By Proposition 3.2, \({{\mathrm{acyc }}}\hbox {Ad}_A\) is the maximum of the geometric multiplicities of the eigenvalues of \(\hbox {Ad}_A\). Those eigenvalues are the banners \(\beta \), and the geometric multiplicity of each \(\beta \) is the total weight with banner \(\beta \). Thus, to prove the lemma we have to show that banner 1 has biggest total weight. \(\square \)

*Remark 4.12*

*Jordan type*of a matrix \(A \in \hbox {Mat}_{d \times d}(\mathbb {C})\) consists on the following data:

- 1.
The number of different eigenvalues.

- 2.
For each eigenvalue, the number of Jordan blocks and their sizes.

### 4.6 Rigidity estimate for c-rectangles

**Lemma 4.13**

In order to prove this lemma, it is convenient to consider separately the cases of non-equatorial and equatorial c-rectangles.

*Proof of Lemma 4.13*

*when*\(\mathtt {C}\)

*is non-equatorial*. For each banner \(\beta \) in \(\mathtt {C}\), let \(n_\beta \) (resp. \(s_\beta \)) be the maximum of \({{\mathrm{rig }}}_+(\hbox {Ad}_A | \mathtt {E}^\square )\) over nonnegative (resp. negative) argument e-rectangles \(\mathtt {E}\) in \(\mathtt {C}\) with banner \(\beta \). For each e-rectangle \(\mathtt {E}\) with banner \(\beta \), choose matrices \(X_{\mathtt {E},1}\), ..., \(X_{\mathtt {E}, n_\beta + s_\beta } \in \mathtt {E}^\square \) such that:

\(\Lambda _\mathtt {E}:= \mathfrak {R}_{\hbox {Ad}_A} ( X_{\mathtt {E},1}, \ldots , X_{\mathtt {E}, m})\) is a transitive subspace of \(\mathtt {E}^\square \);

if \(\mathtt {E}\) has negative argument then \(X_{1} = X_{2} = \cdots = X_{n_\beta } = 0\);

if \(\mathtt {E}\) has nonnegative argument then \(X_{n_\beta +1} = \cdots = X_{n_\beta +s_\beta } = 0\).

So the lemma is proved for non-equatorial \(\mathtt {C}\).

We now consider equatorial c-rectangles. There is a special kind of c-rectangle for which the proof of the rigidity estimate has to follow a different strategy. A c-rectangle is called *exceptional* if it has only the banners 1 and \(-1\) (so it is equatorial and has 4 e-rectangles), each e-rectangle has a single j-rectangle, and all j-rectangles have the same weight.

*Proof of Lemma 4.13*

*when*\(\mathtt {C}\)*is equatorial non-exceptional.* As in the previous case, let \(n_\beta \) (resp. \(s_\beta \)) be the maximum of \({{\mathrm{rig }}}_+(\hbox {Ad}_A|\mathtt {E}^\square )\) over the nonnegative (resp. negative) argument e-rectangles \(\mathtt {E}\) in \(\mathtt {C}\) with banner \(\beta \).

\(\Lambda _\mathtt {E}:= \mathfrak {R}_{\hbox {Ad}_A} (X_{\mathtt {E}, 1}, \ldots , X_{\mathtt {E}, M})\) is a transitive subspace of \(\mathtt {E}^\square \);

\(X_{\mathtt {E}, M} = 0\) if \(\mathtt {E}\) is non-equatorial;

\(X_{\mathtt {E}, M}\) is the identity in \(\mathtt {E}^\square \) if \(\mathtt {E}\) is equatorial.

Now assume by contradiction that (4.9) does not hold. Then we must have equality in (4.13). By what we have just seen, both e-rectangles \(\mathtt {E}_+\) and \(\mathtt {E}_-\) above exist. Then the inequalities in (4.10)–(4.12) become equalities. Since (4.12) is an equality, there must be exactly two equatorial e-rectangles in \(\mathtt {C}\). So the non-equatorial banner \(\beta \) satisfies \(\beta ^{-1} = \beta \), that is, \(\beta =-1\). Since (4.11) is an equality, it follows from Lemma 4.9 that both non-equatorial e-rectangles have the same number of j-rectangles in each column and each row. So there is some \(\ell \) such that all four e-rectangles in \(\mathtt {C}\) have \(\ell \) rows of j-rectangles and \(\ell \) columns of j-rectangles. Since (4.10) is an equality, Lemma 4.8 implies that \(\ell =1\). That is, \(\mathtt {C}\) is a exceptional c-rectangle, a situation which we excluded a priori. This contradiction proves (4.9) and Lemma 4.13 in the present case. \(\square \)

*Proof of Lemma 4.13*

*when*\(\mathtt {C}\)

*is exceptional*. If \(\mathtt {C}\) is exceptional then it has size \(2k \times 2k\) for some \(k\), and the operator \(\hbox {Ad}_A | \mathtt {C}^\square \) is given by \(X \mapsto \hbox {Ad}_L(X)\), where

### 4.7 The final rigidity estimate

Let \(c = c(A)\) be the number of equivalence classes mod \(T\) of eigenvalues of \(A\).

**Lemma 4.14**

*Proof*

\(\Lambda _\mathtt {C}:= \mathfrak {R}_{\hbox {Ad}_A} (X_{\mathtt {C}, 1}, \ldots , X_{\mathtt {C}, m})\) is a transitive subspace of \(\mathtt {C}^\square \);

\(X_{\mathtt {C}, m} = 0\) if \(\mathtt {C}\) is non-equatorial;

\(X_{\mathtt {C}, m}\) is the identity in \(\mathtt {C}^\square \) if \(\mathtt {C}\) is equatorial.

To conclude the proof, we have to show estimate (4.15). First consider a equatorial c-rectangle \(\mathtt {C}\). Since there are \(c\) equatorial c-rectangles, and each of them has a nonzero \({{\mathrm{wgt }}}_1\) value, we conclude that \(r(\mathtt {C}) \leqslant m\), as claimed.

This proves (4.15) and hence Lemma 4.14.

*Example 4.15*

If \(A\) is the matrix of Example 4.1, then Lemma 4.14 gives the estimate \({{\mathrm{rig }}}_+ \hbox {Ad}_A \leqslant 28\). A more careful analysis (going through the proofs of the lemmas) would give \({{\mathrm{rig }}}_+ \hbox {Ad}_A \leqslant 7\) (see Example 4.7). \(\square \)

*Proof of part 2 of Theorem 3.6*

Apply Lemmas 4.14 and 4.11.

## 5 Proof of the hard part of the codimension \(m\) theorem

We showed in Proposition 2.9 that \({{\mathrm{codim }}}\mathcal {P}_m^{(\mathbb {K})} \leqslant m\). In this section, we will prove the reverse inequalities. More precisely, we will first prove Theorem 1.5 and then deduce Theorem 1.4 from it.

### 5.1 Preliminaries on elementary algebraic geometry

#### 5.1.1 Quasiprojective varieties

An algebraic subset of \(\mathbb {C}^n\) is also called an *affine variety*. A *projective variety* is a subset of \(\mathbb {C}\hbox {P}^n\) that can be expressed as the zero set of a family of homogeneous polynomials in \(n+1\) variables. The *Zariski topology* on an (affine or projective) variety \(X\) is the topology whose closed sets are the (affine or projective) subvarieties of \(X\).

An open subset \(U\) of a projective variety \(X\) is called a *quasiprojective variety*. We consider in \(U\) the induced Zariski topology. The affine space \(\mathbb {C}^n\) can be identified with a quasiprojective variety, namely its image under the embedding \((z_1, \ldots , z_n) \mapsto (1: z_1 : \cdots : z_n)\).

If \(X\) and \(Y\) are quasi-projective varieties, then the product \(X \times Y\) can be identified with a quasiprojective variety, namely its image under the Segre embedding; see [12, § 5.1].

Recall the following property from [12, p. 58]:

**Proposition 5.1**

If \(X\) is a projective variety and \(Y\) is a quasiprojective variety, then the projection \(p :X \times Y \rightarrow Y\) takes Zariski closed sets to Zariski closed sets.

A quasiprojective variety is called *irreducible* if it cannot be written as a nontrivial union of two quasiprojective varieties (that is, none contains the other).

#### 5.1.2 Dimension

The dimension \(\dim X\) of an irreducible quasiprojective variety \(X\) may be defined in various equivalent ways (see for instance [8, p. 133ff]). It will be sufficient for us to know that there exists an (intrinsically defined) subvariety \(Y\) of the *singular points of*\(X\) such that in a neighborhood of each point of \(X {\backslash } Y\), the set \(X\) is a complex submanifold of dimension (in the classical sense of differential geometry) \(\dim X\); moreover, each irreducible component of \(Y\) has dimension strictly less than \(\dim X\).

The dimension of a general quasiprojective variety is by definition the maximum of the dimensions of the irreducible components.

The following lemma is useful to estimate the codimension of an algebraic set \(X\) from information about the fibers of a certain projection \(\pi :X \rightarrow Y\).

**Lemma 5.2**

- 1.For each \(j \geqslant 0\), the setis algebraically closed in \(Y\).$$\begin{aligned} C_j = \{ y \in \pi (X) ; \; {{\mathrm{codim }}}\pi ^{-1}(y) \leqslant j \} \end{aligned}$$
- 2.The dimension of \(X\) is given in terms of the dimensions of the \(C_j\)’s by:$$\begin{aligned} {{\mathrm{codim }}}X = \min \limits _{j ; \; C_j \ne \varnothing } \big ( j + {{\mathrm{codim }}}C_j \big ) . \end{aligned}$$(5.1)

In the above, the codimensions of \(\pi ^{-1}(Y)\), \(X\) and \(C_j\) are taken with respect to \(\mathbb {C}\hbox {P}^n\), \(Y \times \mathbb {C}\hbox {P}^n\) and \(Y\), respectively. The proof of the lemma is given in Appendix B (ESM).

*Remark 5.3*

Lemma 5.2 works with the same statement if \(\mathbb {C}\hbox {P}^{n}\) is replaced by \(\mathbb {C}^{n+1}\), provided one assumes that \(X \subset Y \times \mathbb {C}^{n+1}\) is homogeneous in the second factor (i.e., \((y,z) \in X\) implies \((y,tz)\in X\) for every \(t\in \mathbb {C}\)). Indeed, this follows from the fact that the projection \(\mathbb {C}^{n+1}{\backslash }\{0\} \rightarrow \mathbb {C}\hbox {P}^{n}\) preserves codimension of homogeneous sets.

#### 5.1.3 Dimension estimates for sets of vector subspaces

*column-invariant*if

**Theorem 5.4**

#### 5.1.4 The real part of an algebraic set

Let \(X\) be an algebraically closed subset of \(\mathbb {C}^n\). The *real part* of \(X\) is defined as \(X \cap \mathbb {R}^n\). This is an algebraically closed subset of \(\mathbb {R}^n\). Indeed, generators of the corresponding ideal \(f_1,\ldots , f_k\) in \(\mathbb {C}[T_1,\ldots , T_n]\) can be replaced by the corresponding real and imaginary parts polynomials.

As in the complex case, there are many equivalent algebraic–geometric definitions of dimensions of real algebraic or semialgebraic sets. We just point out that a real algebraic or semialgebraic set admits a stratification into real manifolds such that the maximal differential–geometric dimension of the strata coincides with the algebraic–geometric dimension (see [2, § 3.4] or [3, p. 50]).

The following is an immediate consequence of [2, Prop. 3.3.2]:

**Proposition 5.5**

If \(X\) is an algebraically closed subset of \(\mathbb {C}^n\), then \(\dim _\mathbb {R}(X \cap \mathbb {R}^n) \leqslant \dim _\mathbb {C}X\).

### 5.2 Rigidity and the dimension of the poor fibers

**Lemma 5.6**

For any \(A \in \hbox {GL}(d,\mathbb {C})\), the codimension of \(\mathcal {P}_m(A)\) in \(\mathfrak {gl}(d,\mathbb {C})^m\) is at least \(m + 1 - r(A)\).

The lemma follows easily from Theorem 5.4 above:

*Proof*

Fix \(A \in \hbox {GL}(d,\mathbb {C})\) and write \(r=r(A)\). We can assume that \(r \leqslant m\), otherwise there is nothing to prove. By definition, there exists a \(r\)-dimensional subspace \(E \subset \mathfrak {gl}(d,\mathbb {C})^m\) such that \(\mathfrak {R}_{\hbox {Ad}_A}(\hbox {Id}\vee E)\) is transitive. Identify \(\mathfrak {gl}(d,\mathbb {C})\) with \(\mathbb {C}^{d^2}\) and thus regard \(\mathcal {P}_m(A)\) as a subset of \(\hbox {Mat}_{d^2 \times m} (\mathbb {C})\). Since the set \(\mathcal {P}_m\) is algebraically closed and saturated (recall § 2.3), the fiber \(\mathcal {P}_m(A)\) is algebraically closed and column-invariant, as required by Theorem 5.4. In the notation (5.2), we have \(E \not \in [\![\mathcal {P}_m(A) ]\!]\). So Theorem 5.4 gives the desired codimension estimate.

### 5.3 How rare is high rigidity?

**Lemma 5.7**

Lemma 5.7 is basically a consequence of Theorem 3.6, using the following construction:

**Lemma 5.8**

- 1.
Each \(\mathcal {G}(A)\) contains \(A\).

- 2.
Each \(\mathcal {G}(A)\) is an immersed manifold of codimension \(a(A)-c(A)\).

- 3.
There are only countably many different sets \(\mathcal {G}(A)\).

*Proof*

*Proof of Lemma 5.7*

If \(k=1\), then \(M_1 = \hbox {GL}(d,\mathbb {C})\) (since \(d \geqslant 2\)), so there is nothing to prove. Consider \(k \geqslant 2\). We have already shown in Sect. 2.3 that \(\mathcal {P}_k\) is algebraic. Since \(M_k = \{ A \in \hbox {GL}(d,\mathbb {C}) ; \; \forall \hat{X} \in \mathfrak {gl}(d,\mathbb {C})^{k},\,(A, \hat{X}) \in \mathcal {P}_k \}\), it is evident that \(M_k\) is algebraically closed as well. We are left to estimate its dimension.

### 5.4 Proof of Theorems 1.5 and 1.4

We apply Lemmas 5.6 and 5.7 to prove one of our major results:

*Proof of Theorem 5.7*

The proof above only used that \({{\mathrm{codim }}}C_j \geqslant m-j\). On the other hand, using the full power of (5.6) we obtain:

**Scholium 5.9**

*Proof*

The projection of \(\mathcal {F}_m\) on \(\hbox {GL}(d,\mathbb {C})\) is \(C_{m-1}\). Use Lemma 5.2 (together with Remark 5.3) and (5.6).

Next, let us consider the real case:

*Proof of Theorem 1.4*

The real part of \(\mathcal {P}^{(\mathbb {C})}_m\) is a real algebraic set which, in view of Proposition 5.5, has codimension at least \(m\). Recall from Sect. 2.3 that this set contains the semialgebraic set \(\mathcal {P}^{(\mathbb {R})}_m\), which therefore has codimension at least \(m\). Since we already knew from Proposition 2.9 that \({{\mathrm{codim }}}\mathcal {P}^{(\mathbb {R})}_m \leqslant m\), the theorem is proved.

## 6 Proof of the main result

We now use Theorem 1.4 and transversality theorems to prove our main result. For precise definitions and statements on the objects used in this section, see Appendix C (ESM).

*stratification*is a filtration by closed subsets of a smooth manifold \(X\)

We say that a \(C^1\)-map is *transverse* to that stratification if it is transverse to each of the submanifolds \(\Gamma _i\). There are explicit, so-called *Whitney conditions* that guarantee that a stratification behaves nicely with respect to transversality, as the next proposition shows. A stratification satisfying those conditions is called a *Whiney stratification*. By the classical Theorem C.1 stated in Appendix C (in ESM) (see for instance [7]), any semi-algebraic subset of an affine space admits a canonical Whitney stratification.

We refer the reader to Appendix C (ESM) for the definitions of jets, jet extensions and for a proof of the following:

**Proposition 6.1**

Let \(X\), \(Y\) be \(C^\infty \)-manifolds without boundary. Let \(\Sigma \) be a Whitney stratified closed subset of the set of 1-jets from \(X\) to \(Y\). Then the set of maps \(f \in C^2(X,Y)\) whose 1-jet extension \(j^1f\) is transverse to \(\Sigma \) is \(C^2\)-open and \(C^\infty \)-dense in \(C^2(X,Y)\) (i.e., its intersection with \(C^r(X,Y)\) is \(C^r\)-dense, for every \(2\leqslant r\leqslant \infty \)).

*Proof of Theorem 1.1*

*rich*if the datum \(\mathbf {A}= (A(u),B_1, \ldots ,B_m)\) is rich, or equivalently, if for sufficiently large \(N\), the input \((u,\ldots ,u)\in \mathcal {U}^N\) is universally regular for the system (1.4). If the jet is not rich then it is called

*poor*.

Applying Proposition 6.1, we obtain a \(C^2\)-open \(C^\infty \)-dense set \(\mathcal {O}\subset C^2(\mathcal {U},\)\(\hbox {GL}(d,\mathbb {C}))\) formed by maps \(A\) that are transverse to the stratification (6.2) of the set of poor jets. Since the codimension of the stratification equals the dimension of \(\mathcal {U}\), if \(A \in \mathcal {O}\), then the points \(u\) for which \(j^1 A(u)\) is poor form a 0-dimensional set. This proves Theorem 1.1.

## Notes

### Acknowledgments

We are grateful for the hospitality of Institute Mittag–Leffler, where this work begun to take form. We thank R. Potrie, L. San Martin, S. Tikhomirov, and C. Tomei for valuable discussions. We thank the referees for corrections, references to the literature, and other suggestions that helped to improve the exposition.

## Supplementary material

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