Journal of Algebraic Combinatorics

, Volume 37, Issue 4, pp 757–776 | Cite as

Gérard–Levelt membranes

Open Access


We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical Gérard–Levelt lattice saturation procedure can be geometrically understood in terms of a projection on the tropical linear space attached to a subset of the local affine Bruhat–Tits building, which we call the Gérard–Levelt membrane. This provides a way to compute the true Poincaré rank, but also the Katz rank of a meromorphic connection without having to perform either gauge transforms or ramifications of the variable. We finally present an efficient algorithm to compute this tropical projection map, generalising Ardila’s method for Bergman fans to the case of the tight-span of a valuated matroid.


Meromorphic connections Tropical convexity Valuated matroids 

1 Introduction

Given a meromorphic linear differential system on the Riemann sphere,
$$ \frac{dX}{dz}=A(z)X\quad\mathrm{with}\ A(z)\in\mathrm{M}_n \bigl(\mathbb{C}(z) \bigr), $$
it is important to determine whether a singularity of A is a regular or an irregular singular point for the system (1). Unlike the case with scalar linear differential equations, for which there is a purely algebraic condition on the orders of the poles of the coefficients due to L. Fuchs [10], a system (1) can display arbitrarily high pole orders at a regular singularity.

Example 1

has a regular singularity at z=0 for any N∈ℤ.
Consider the differential system, expanded in the neighbourhood of the singular point (assumed for simplicity to be z=0) under the following form, where we put \(\theta=z\frac{d}{dz}\):
$$ \theta X=\frac{1}{z^{p}}\sum_{i\geq0}A_iz^iX \quad\mathrm{with}\ p\geq0\ \mathrm{and}\ A_0\neq0\ \mathrm{if}\ p>0. $$
The integer p is known as the Poincaré rank\(\mathfrak{p}(A)\) of the system. Finding the type of singularity involves knowing the minimum value m(A) of this rank, sometimes known as the true Poincaré rank, under gauge transformations
$$ A_{[P]}=P^{-1}AP-P^{-1}\theta P\quad\mathrm{with}\ P= \sum_{k\geq k_0}P_kz^k\in \mathrm{GL}_n(K) \ \mathrm{where}\ K=\mathbb{C} \bigl((z) \bigr), $$
in the sense that z=0 is a regular singularity of (1) if and only if m(A)=0.

Several lines of research have been opened to tackle this problem. The most classical tries to iteratively construct a suitable gauge transformation P, usually coefficient by coefficient in its series expansion. Featured methods rely on the linear algebra over ℂ involved by (4), like Moser and continuators [4, 15, 21], whose methods are widely used nowadays in computer algebra, or other researchers such as [3, 13], while [2] uses Lie group theoretic tools.

The nature of a singularity of A can also be considered from the point of view of meromorphic connections [7], and especially, as a question of stability of certain lattices under the differential operator induced by the connection [14, 20]. We focus here specifically on the approach of saturating lattices used by Gérard and Levelt [12]: the true Poincaré rank is the minimum integer k such that the sequence of k-saturated lattices (recalled in Sect. 2.1) eventually stabilises.

Recent work has shown close relations between the geometric framework of the Bruhat–Tits building of SL(K), for some discrete-valued field such as K=ℂ((z)), and tropical convexity [16, 17, 25]. In particular, any finite union of apartments in the Bruhat–Tits building (a so-called membrane) can be faithfully represented as the set of integer-valued points of the tropical linear space defined by a tropical Plücker vector (or valuated matroid). If a membrane \(\mathcal{M}\) is generated by vectors v1,…,vm, a lattice Λ in \(\mathcal{M}\) admits as non-unique representative vector any u∈ℤm such that \(\varLambda=\sum_{i=1}^{m}\mathcal{O}z^{-u_{i}}v_{i}\), where \(\mathcal{O}\) is the valuation ring of K. Results of Keel and Tevelev [17] show that, when lattices are in a same membrane, they are homothetic if and only if their representative points are projected on the same point of the attached tropical linear space by an explicit nearest point projection map ([16], see also [5, 11]).

We show here that Gérard and Levelt’s approach can be formulated and efficiently computed in this framework. Let ∇ be a meromorphic connection acting on an n-dimensional vector space V over K. We construct first the Gérard–Levelt membrane\(\mathcal{M}_{\varLambda}\) that contains all the relevant k-saturated lattices of a given lattice Λ (Proposition 8). Our main result is the following tropical version of Gérard–Levelt’s lattice stabilisation criterion. Let Λ be a lattice, let \(u^{\ell}_{k}\) represent the th order k-saturated lattice of Λ, let πΛ be the projection map onto the attached tropical linear space LΛ, and
$$I_{\varLambda}= \bigl\{k\in\mathbb{R}^+\,|\,\pi_{\varLambda } \bigl(u^{n}_k \bigr)=\pi_{\varLambda} \bigl(u^{n-1}_k \bigr) \bigr\}. $$

Theorem 1

The true Poincaré rankmand the Katz rankκof a connectionacting onVsatisfy
$$m=\min I_{\varLambda}\cap\mathbb{N}\quad\mbox{\textit{and}}\quad \kappa=\min I_{\varLambda}\quad \mbox{\textit{for any lattice}}\ \varLambda\ \mbox{\textit{in}}\ V. $$

It is remarkable that same formula holds for the computation of the true Poincaré rank of the connection, and for a more subtle invariant like the Katz rank, which can moreover be computed without having to either compute a single gauge transformation or perform the usually required ramification of the variable.

Example 2


Let dX/dz=AX with One gets \(\pi_{\varLambda}(u^{4}_{k})=\pi_{\varLambda}(u^{3}_{k})\) if and only if \(k\geq\frac{3}{2}\), hence m=2 and \(\kappa=\frac{3}{2}\).

Finally, we give in Sect. 5 an efficient algorithm to compute this projection map. Indeed, the explicit algorithms given in [16] are too complex in practice. We generalise the algorithmic approach to tropical projection developed by Ardila [1], and further by Rincón [23], for ordinary matroids, to the case of valuated matroids, defined by Dress and collaborators [9]. Namely, if p is a valuated matroid of rank n on [m], and Lp is the tropical linear space [24] attached to it, then we have the following result.

Theorem 2

Letx∈ℝmand letBbe anx-minimal base ofp. Then\(\omega=\pi_{L_{p}}(x)\)can be computed in the following way:
$$\omega_i=\left \{\begin{array}{l@{\quad}l} x_i & \mbox{\textit{if}}\ i\in B,\\[3pt] \min_{u\neq i}(p(B\cup\{i\}\backslash\{u\})-p(B)+x_u) & \mbox{\textit{otherwise}}. \end{array} \right . $$

The algorithm based on this result,1 which computes the nearest -projection on the tight-span of a valuated matroid, has a wider applicability than the differential computations explained in the previous parts, especially in phylogenetics [8].

2 Meromorphic connections

A meromorphic connection is a map \(\nabla:V\simeq K^{n} \rightarrow\varOmega(V)=V\otimes_{K}\varOmega^{1}_{\mathbb{C}}(K)\) which is ℂ-linear and satisfies the Leibniz rule
$$\nabla(fv)=v\otimes df+f\nabla v\quad \mathrm{for}\ f \in K\ \mathrm{and}\ v \in V. $$
The matrix Mat(∇,(e)) is given by \(\nabla e_{j}=-\sum^{n}_{i=1} e_{i}\otimes\varOmega_{ij}\) for a basis (e). A basis change P∈GLn(K) gauge-transforms the matrix of ∇ by
$$ \varOmega_{[P]}=P^{-1}\varOmega P-P^{-1}\,dP. $$
Contracting with \(z^{k+1}\frac{d}{dz}\) yields a differential operator ∇k, and system (1) is the expression of ∇−1(v)=0 in the basis (e).
A latticeΛ in V is a free sub-\(\mathcal{O}\)-module of rank n, that is, a module of the form
$$\varLambda=\bigoplus_{i=1}^n\mathcal{O}e_i \quad \mathrm{for\ some\ basis}\ (e)\ \mathrm{of}\ V. $$
The lattice Λ induces a valuation on V, defined by
$$v_{\varLambda}(x)=\max \bigl\{k\in\mathbb{Z}\,|\,x\in z^k \varLambda \bigr\}. $$
Let Λ be the set of all lattices in V. The Poincaré rank ofon the latticeΛ is defined as the integer and the true Poincaré rank as \(m(\nabla)=\min_{\varLambda \in \boldsymbol{\varLambda}} \mathfrak{p}_{\varLambda }(\nabla)\).

2.1 Gérard–Levelt’s saturated lattices

For any vector eV and any derivation τ∈Der(K/ℂ), define for ∈ℕ the family
$$Z_{\tau}^{\ell}(e)= \bigl(e,\nabla_{\tau}e,\ldots, \nabla_{\tau }^{\ell} e \bigr). $$
The module \(\mathcal{O}_{\tau}^{\ell}(e)\) induced over \(\mathcal{O}\) by \(Z_{\tau}^{\ell }(e)\) only depends on the valuation v(τ) of the derivation τ. We can therefore restrict to the particular derivations \(\tau_{k}=z^{k+1}\frac{d}{dz}\) for k∈ℕ. In this case, we put \(\nabla_{\tau_{k}}=\nabla_{k}\), and denote by \(Z^{\ell}_{k}(e)\) and \(\mathcal{O}^{\ell}_{k}(e)\) the corresponding objects. For k≥1, Gérard and Levelt define the lattices (see also [19])
$$F^{\ell}_k(\varLambda)=\varLambda+\nabla_k \varLambda+\cdots+ \nabla_k^{\ell}\varLambda. $$
Note that
$$F^{\ell}_k(\varLambda)=\sum_{i=1}^n Z^{\ell}_k(e_i)\quad\mathrm{for\ any\ basis}\ (e_1,\ldots,e_n)\ \mathrm{of}\ \varLambda. $$

Theorem 3

(Gérard, Levelt)

The true Poincaré rankm(∇) ofis
$$m(\nabla)=\min \bigl\{k\in\mathbb{N}\,|\,F^{n}_k( \varLambda)=F^{n-1}_k( \varLambda) \bigr\}\quad\mbox{ \textit{for any lattice}}\ \varLambda \subset V. $$
This means that km(∇) if and only if, for some (equivalently, any) lattice Λ in V, the Poincaré rank on \(F^{n-1}_{k}(\varLambda)\) is at most k. Stated otherwise, finding the true Poincaré rank is finding the largest lattice whose Poincaré rank is bounded by its index in the following sequence:
$$ F^{n-1}_0(\varLambda)\supset\cdots\supset F^{n-1}_{p-1}( \varLambda) \supset\varLambda. $$
Let us extend this notation to multi-indices. Let ≥0, and let α=(α1,…,α)∈ℤ be an integer multi-index of length |α|= and weightw(α)=α1+⋯+α. Let us define also the partial multi-indices α|j=(α1,…,αj) and
$$\nabla^{\alpha}=\nabla_{\alpha_{\ell}}\circ\cdots\circ \nabla_{\alpha_1}. $$
Let by convention α|0=ϵ and ∇ϵ=idV for the empty sequence ϵ. Let finally \(\mathcal{O}^{\alpha}(e)\) be the \(\mathcal{O}\)-module spanned by the sequence
$$Z^{\alpha}(e)= \bigl(\nabla^{\alpha_{\vert j}}e \bigr)_{0\leq j\leq {\alpha}} . $$

Lemma 3

For anyα=(α1,…,α)∈ℤ, one has
$$\nabla^{\alpha}=z^{w(\alpha)}P_{\alpha}(\nabla_0) \quad\mbox {\textit {where}}\ P_{\alpha}(X)=X(X+\alpha_1)\cdots \bigl(X+w(\alpha_{\vert\ell-1})\bigr)\in \mathbb{Z}[X]. $$


The proof goes by induction on the length of the multi-index α. Let D=∇0. The claim obviously holds for a multi-index of length 0, with Pϵ=1, so assume that there exists Pα∈ℤ[X] such that ∇α=zw(α)Pα(D) for α. Let β∈ℤ+1. Then by definition, we have Indeed, \(P_{\beta_{\vert\ell}}(D)\) commutes with D since it has by assumption constant coefficients. The result follows, since we have then \(P_{\beta}(X)=P_{\beta_{\vert\ell}}(X)(X+w(\beta_{\vert\ell}))\). □

Lemma 4

LetΛbe a lattice inV. For any∈ℕ andα∈ℕ, the\(\mathcal{O}\)-module\(\mathcal{O}^{\alpha}(e)\)is spanned over\(\mathcal{O}\)by the family
$$\bigl(e,z^{\alpha_1}\nabla_0 e,\ldots,z^{w(\alpha_{\vert\ell-1})} \nabla_0^{\ell-1} e \bigr). $$


According to Lemma 3, the family \(Z^{\alpha_{\vert\ell -1}}(e)\) is related to \(Z_{0}^{\ell-1}(e)\) by the matrix \(P=Az^{W_{\alpha}}\) where
$$W_{\alpha}=\mathrm{diag}\bigl(0,\alpha_1,\ldots,w( \alpha_{\vert\ell-1})\bigr), $$
and A is an upper triangular integer matrix with diagonal entries equal to 1, therefore \(A\in\mathrm{SL}_{\ell}(\mathbb{Z})\subset\mathrm {GL}_{\ell }(\mathcal{O})\). The families \(z^{W_{\alpha}}Z_{0}^{\ell-1}(e)\) and \(Z^{{\alpha}_{\vert\ell-1}}(e)\) are related by the matrix
$$\tilde{P}=z^{-W_{\alpha}}P=z^{-W_{\alpha}}Az^{W_{\alpha}}\quad \mbox{whose entries are}\ A_{ij}z^{w(\alpha_{\vert j})-w(\alpha_{\vert i})}. $$
Since A is upper triangular, and the partial sums αi+⋯+αj are non-negative, the matrix \(\tilde{P}\) is in \(\mathrm {GL}_{n}(\mathcal{O})\), and therefore both families span the same \(\mathcal{O}\)-module. □

3 Tropical convexity and lattices

Let M={d1,…,dm} be lines in V such that d1+⋯+dm=V, and consider the subset of Λ defined by
$$[M]=\{\ell_1+\cdots+\ell_m \,|\,\ell_i \ \mathrm{is\ a\ lattice\ in}\ d_i\}. $$
Following Keel and Tevelev, who call in [17] the set induced by [M] modulo homothety a membrane, we call this the affine membrane spanned byM.
For a choice \(\mathcal{A}=(v_{1},\ldots,v_{m})\) of non-zero vectors in the lines di, any lattice in the membrane defined by M={d1,…,dm} can be represented (non uniquely) by an integer-valued point as follows: a lattice point u∈ℤm corresponds to the lattice
$$ \varLambda_u=\sum_{i=1}^m \mathcal{O} z^{-u_i}v_i. $$

Membranes spanned by m lines in the Bruhat–Tits building have a faithful representation as tropical linear spaces in m-dimensional space.

Let (ℝ=ℝ∪{∞},⊕,⊙) be the tropical semialgebra, with the operations
$$x\oplus y=\min(x,y)\quad \mathrm{and}\quad x\odot y=x+y \quad \mathrm{for}\ x,y\in \mathbb{R}_{\infty}. $$
An affine membrane M and a basis (e) of V determine a valuated matroid
$$ \begin{array}{rcl} {p:}{\: [m]^n }& \rightarrow& {\mathbb{R}_{\infty}} \\[6pt] {\omega} &\mapsto& {v(\det_{(e)} M_{\omega})} \end{array} $$
where \(M_{\omega}=(v_{\omega_{1}},\ldots,v_{\omega_{n}})\) is the subfamily of vectors of M indexed by ω. To a valuated matroid p of rank n over [m] there is a tropical linear space \(L_{p}\subset\mathbb{R}_{\infty}^{m}\) attached as follows:
$$ L_p= \Bigl\{x\in\mathbb{R}_{\infty}^m \,\Big\vert\, \forall\tau\in{{[m]} \choose{n+1}},\:\min_{1\leq i\leq n+1}p\bigl(\tau\backslash\{ \tau_i\} \bigr)+x_{\tau_i}\ \mathrm{is\ attained\ twice} \Bigr\}. $$
Depending on the authors, Lp is said to be a tropical convex cone in \(\mathbb{R}_{\infty}^{m}\) [5] or a tropical polytope [16] in \(\mathbb{R}_{\infty}^{m}/\mathbb{R} (1,\ldots,1)\). Both definitions mean that
$$\lambda\odot u\oplus\mu\odot v\in L_p\quad \mathrm{for\ any}\ \lambda,\mu \in\mathbb{R}_{\infty}\ \mathrm{and}\ u,v\in L_p. $$

Example 5

Take the membrane generated by the four vectors
$$v_1=\left ( \begin{matrix} 1\\ 0 \end{matrix} \right ),\qquad v_2=\left ( \begin{matrix} 0\\ 1 \end{matrix} \right ),\qquad v_3=\left ( \begin{matrix} z^{-N}\\ 1 \end{matrix} \right )\quad \mathrm{and}\quad v_4=\left ( \begin{matrix} \alpha z^{-N}\\ 1 \end{matrix} \right )\quad \mathrm{with}\ \alpha\neq1. $$
Applying directly formula (11) to the valuated matroid p defined in (10) readily shows that Lp consists of four half-planes in \(\mathbb{R}_{\infty}^{4}\) defined by with P=(0,−N,−N,−N). These four half-planes intersect along the line P+ℝ(1,1,1,1).
According to [5, 11, 16], the tropical nearest point projection map\(\pi_{L_{p}}: \mathbb{R}_{\infty}^{m} \rightarrow L_{p}\) is defined by
$$ \pi_{L_p}(x)=\min\{w\in L_p \,|\,w\geq x\}, $$
where the minimum is taken for the coordinate-wise ordering
$$y\geq x\quad \iff\quad y_i\geq x_i\quad \mathrm{for\ all}\ 1\leq i \leq m. $$
There are at least two other ways, given by the authors of [16], adapting [1], to characterise or compute \(\pi_{L_{p}}(x)\).
Blue Rule:
\(\pi_{L_{p}}(x)=(w_{1},\ldots,w_{m})\) with
$$w_i=\min_{\sigma\in{{[m]}\choose{n-1}}}\max_{j\neq \sigma} \bigl(p\bigl(\sigma\cup \{i\}\bigr)-p\bigl( \sigma\cup\{j\}\bigr)+x_j \bigr). $$
Red Rule:

Starting with v=(0,…,0)∈ℝm, for every \(\tau\in{{[m]}\choose{n+1}}\) such that \(\alpha=\min_{1\leq i\leq n+1}p(\tau\backslash\{\tau_{i}\})+x_{\tau_{i}}\) is only attained once, say at τi, compute γ=βα where β is the second smallest number in that collection, and put \(v_{\tau_{i}}:=\max(v_{\tau_{i}},\gamma)\). Then \(\pi_{L_{p}}(x)=x+v\).

Theorem 4

The map\(\pi_{L_{p}}:\mathbb{R}_{\infty}^{m} \rightarrow L_{p}\)induces a bijectionΨMbetween [M] and the lattice points inLp
$$\varPsi_M(\varLambda_u) = \pi_{L_p}(u_1, \ldots,u_m). $$

If we let \(z^{-u}v=(z^{-u_{1}}v_{1},\ldots,z^{-u_{m}}v_{m})\), then zuv and zuv span the same lattice Λ if and only if \(\pi_{L_{p}}(u)=\pi_{L_{p}}(u^{\prime})\).


Let \(\varLambda=\sum_{i=1}^{m} \mathcal{O}z^{-u_{i}}v_{i}\), and let \(w=\pi_{L_{p}}(u)\). According to [17], Theorem 4.17 (see also [16], Theorem 18), there exists α∈ℝ such that wi=vΛ(vi)+α for all 1≤im. By definition, vΛ(x)=max{k∈ℤ | xzkΛ}. Accordingly, we have \(z^{-w_{i}}v_{i}\in z^{-\alpha }\varLambda\), and thus α≥0. By (12), we get α=0 and thus
$$ \pi_{L_p}(u)= \bigl(v_{\varLambda}(v_1), \ldots,v_{\varLambda}(v_m) \bigr). $$
By construction, if u′≥u, then \(z^{-u_{i}}v_{i}=z^{(u'_{i}-u_{i})}z^{-u'_{i}}v_{i}\in\mathcal{O}z^{-u'_{i}}v_{i}\), for 1≤im, hence ΛΛu. Since in particular wu holds, we get ΛΛw. Conversely, we have \(\varLambda_{w}=\sum_{i=1}^{m} \mathcal{O}z^{-w_{i}}v_{i}\subset\varLambda\). Therefore, if \(\pi_{L_{p}}(u)=\pi_{L_{p}}(u')\) then Λu=Λu. The converse follows directly from (13). □

As mentioned in [16], as soon as the projection \(\pi_{L_{p}}(u)\) is computed, one can also determine a basis of the lattice Λu.

Lemma 6

LetM={v1,…,vm} be a set of vectors of rankninV, and letLpbe the associated tropical linear space. Let\((w_{1},\ldots,w_{m})=\pi_{L_{p}}(u)\). For anyn-subsetτ⊂{1,…,m} such that\(p(\tau)-w_{\tau_{1}}-\cdots-w_{\tau_{n}}\)is minimal, the subfamily\((v_{\tau_{1}},\allowbreak \ldots,v_{\tau_{n}})\)is an\(\mathcal{O}\)-basis ofΛu.


Fix a basis (e) in V. Let M be the n×m matrix of coordinates of v1,…,vm in (e), and let Mτ denote the square submatrix obtained by selecting the columns having their index in τ. The family
$$z^{-u_{\tau}}v_{\tau} = \bigl(z^{-u_{\tau_1}} v_{\tau_1}, \ldots,z^{-u_{\tau_n}}v_{\tau_n} \bigr) $$
spans an \(\mathcal{O}\)-submodule Λτ of Λw. If the rank of the subfamily \(v_{\tau }=(v_{\tau_{1}},\ldots ,v_{\tau_{n}})\) is not n, then we have \(v(\det_{(e)}(M_{\tau}z^{-u_{\tau}}))=\infty\). Let us therefore choose two n-subsets τ and τ′ of [m], and assume that both vτ and vτ have full rank. The matrix of the basis change from \(z^{-u_{\tau}}v_{\tau}\) to \(z^{-u_{\tau^{\prime}}}v_{\tau^{\prime}}\) is given by \(P=z^{u_{\tau^{\prime}}}M_{\tau^{\prime}}^{-1}M_{\tau}z^{-u_{\tau }}\). Therefore, we have ΛτΛτ if and only if v(P)≥0, that is,
$$p(\tau)-w_{\tau_1}-\cdots-w_{\tau_n}\geq p \bigl( \tau^{\prime} \bigr)-w_{\tau^{\prime}_1}-\cdots-w_{\tau^{\prime}_n}. $$
By assumption, there exists a subset τ such that Λτ=Λw. Since Λw is the largest of all the submodules Λτ, the result follows. □

As a consequence, one can find a basis of the lattice Λu by computing the minimum of a valuated matroid, which can be performed efficiently by a greedy algorithm (see Algorithms 2 and 3 in Sect. 5).

Example 7

Continuing Example 5, consider the hyperplanes H0={xy=N}, H1={xz=N}, H2={xt=N}, H3={yz=0}, H4={yt=0} and H5={zt=0}. Let \(H_{i}^{+}\) (resp. \(H_{i}^{-}\)) be the half-space defined by replacing equality by ≥ (resp. ≤) in the defining equations of hyperplane Hi. Then \(\mathbb{R}_{\infty}^{4}\) is subdivided into the fiber subsets \(R_{i}=\pi_{L_{p}}^{-1}(L_{i})\) defined by To get the explicit projection formulæ, one must subdivide further the regions Ri into three regions each. The following table sums up the properties of the projection map \(\pi_{L_{p}}\), with the convention that, e.g. for row one: if \(u=(x,y,z,t)\in H_{0}^{+}\cap H_{1}^{+}\cap H_{2}^{-}\cap H_{4}^{-} \cap H_{5}^{-}\), then \(\pi_{L_{p}}(u)=(x,x-N,x-N,t)\in\varDelta_{1}\). The underlined coordinates correspond to the u-minimal base (as explained in Lemma 6).
$$\arraycolsep=5pt \begin{array}{@{}llllllll@{}} \hline H_0 & H_1 & H_2 & H_3 & H_4 & H_5 & \pi_{L_p}(x,y,z,t) & \varDelta_i \\ \hline + & + & - & & - & - & (\underline{x},x-N,x-N,\underline{t}) & \varDelta_1 \\ - & & - & + & - & - & (y+N,\underline{y},y,\underline{t}) & \varDelta_1 \\ & - & - & - & - & - & (z+N,z,\underline{z},\underline{t}) & \varDelta_1 \\[6pt] + & - & + & - & & + & (\underline{x},x-N,\underline{z},x-N) & \varDelta_2 \\ - & - & & - & + & + & (y+N,\underline{y},\underline{z},y) & \varDelta_2 \\ & - & - & - & - & + & (t+N,t,\underline{z},\underline{t}) & \varDelta_2 \\[6pt] - & + & + & + & + & & (\underline{x},\underline{y},x-N,x-N) & \varDelta_3 \\ - & - & & + & + & + & (z+N,\underline{y},\underline{z},z) & \varDelta_3 \\ - & & - & + & + & - & (t+N,\underline{y},t,\underline{t}) & \varDelta_3 \\[6pt] + & + & + & + & + & & (\underline{x},\underline{y},y,y) & \varDelta_4 \\ + & + & + & - & & + &(\underline{x},z,\underline{z},z) & \varDelta_4 \\ + & + & + & & - & - & (\underline{x},t,t,\underline{t}) & \varDelta_4 \\ \hline \end{array} $$

4 The Gérard–Levelt membranes

Proposition 8

Fix a basis (e) ofΛ, and≥0. Let [M] be the membrane spanned by the vectors\((\nabla_{0}^{j} e_{i})_{1\leq i \leq n,0\leq j \leq\ell}\). Then\(F^{\ell'}_{k}(\varLambda)\in[M_{\ell}]\)for allk≥0 and′≤.


For the considered basis (e), the lattice \(L=F^{\ell'}_{k}(\varLambda )\) satisfies
$$L=\mathcal{O}^{\alpha}(e_1)+\cdots+\mathcal{O}^{\alpha}(e_n) \quad \mathrm{with}\ \alpha= \bigl(0,k,\ldots,k \ell' \bigr). $$
Reordering terms as \((e_{1},\ldots,e_{n},\nabla_{0} e_{1},\ldots\nabla_{0} e_{n},\ldots,\nabla_{0}^{\ell'} e_{n})\), formula (9) and Lemma 4 imply that L can be represented in the membrane [M] by the lattice point
$$\bigl(\underbrace{0,\ldots,0}_{n\ \mathrm{times}},\underbrace{-k, \ldots,-k}_{n\ \mathrm{times}},\ldots,\underbrace{-k\ell',\ldots,-k \ell'}_{n\ \mathrm{times}} \bigr). $$
Since by definition, \(z^{-v_{\varLambda}(v)}v\in\varLambda\) holds for any vV, the module L can also be represented as an element of the membrane [M] by
$$\bigl(\underbrace{0,\ldots,0}_{n\ \mathrm{times}}, \underbrace{-k, \ldots,-k}_{n\ \mathrm{ times}},\ldots,\underbrace{-k\ell',\ldots,-k \ell'}_{n\ \mathrm{times}}, v_{\varLambda} \bigl(\nabla_0^{\ell'+1} e_1 \bigr),\ldots,v_{\varLambda} \bigl(\nabla_0^{\ell} e_n \bigr) \bigr). $$

The lattices \(F^{\ell}_{k}(\varLambda)\) for 0≤n can therefore all be seen as elements of the same membrane [Mn].

Definition 9

\(\mathcal{M}_{\varLambda}=[M_{n}]\)is called the Gérard–Levelt membrane attached toΛ. For any basis (e), the lattice\(F^{\ell}_{k}(\varLambda)\)is represented by the lattice point
$$u^{\ell}_k= \bigl(\underbrace{0,\ldots,0}_{n\ \mathrm{times}}, \underbrace{-k,\ldots,-k}_{n\ \mathrm{times}},\ldots,\underbrace {-k\ell,\ldots,-k \ell}_{n\ \mathrm{times}}, v_{\varLambda} \bigl(\nabla_0^{\ell+1} e_1 \bigr),\ldots,v_{\varLambda} \bigl(\nabla_0^{n} e_n \bigr) \bigr). $$
If Mat(∇0,(e))=A for a basis (e) of Λ, then \(\mathcal{M}_{\varLambda}\) is described in (e) by the n×n(n+1) saturation matrix
$$ \mathbf{M}=\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} I_n & A & \ldots& A_n \end{array} \right )\quad \mathrm{where}\ A_{k+1}= \biggl(z\frac{d}{dz}+A \biggr)A_k \ \mathrm{and}\ A_0=I_n. $$
The tropical projection πΛ onto the tropical linear space LΛ attached to the Gérard–Levelt membrane \(\mathcal{M}_{\varLambda}\) maps a point u to a unique representative. Checking if km(∇) requires to know if the lattice points \(u^{n-1}_{k}\) and \(u^{n}_{k}\) represent the same lattice, that is,
$$\pi_{\varLambda} \bigl(u^{n}_k \bigr)= \pi_{\varLambda} \bigl(u^{n-1}_k \bigr). $$

Corollary 10

For anyΛ, we have\(m(\nabla)=\min\{k\in\mathbb{N}\,|\,\pi_{\varLambda }(u^{n}_{k})=\pi_{\varLambda}(u^{n-1}_{k})\}\).

Example 11

For Example 1, the saturation matrix is The lattice \(F^{2}_{k}(\mathcal{O}^{2})\) is represented by the point \(u^{2}_{k}=(0,0,-k,-k,-2k,-2k)\) and \(F^{1}_{k}(\mathcal{O}^{2})\) by \(u^{1}_{k}=(0,0,-k,-k,0,-N+1)\). However, the membrane can be reduced to dimension 4, and seen as a particular case of Example 5, with generators
$$v_1=\left ( \begin{matrix} 1\\ 0 \end{matrix} \right ),\qquad v_2=\left ( \begin{matrix} 0\\ 1 \end{matrix} \right ),\qquad v_3=\left ( \begin{matrix} z^{-N+1}\\ 1 \end{matrix} \right ) \quad \mathrm{and}\quad v_4=\left ( \begin{matrix} (3-N)z^{-N+1} \\ 1 \end{matrix} \right ) $$
and the lattice representatives can be replaced by \(u^{2}_{k}=(0,0,-k,-2k)\) and \(u^{1}_{k}=(0,0,-k,-N+1)\). According to the projection formulæ in Example 7, we get therefore we get m(A)=0. The lattices represented by the points correspond to \(\varLambda_{1}=\mathcal{O}^{2}\) in the case (15) and \(\varLambda_{2}=\mathcal{O} v_{2}\oplus z^{k} v_{3}\) in the case (16).

4.1 Tropical computation of the Katz rank

It is well known (cf. [3]) that in a neighbourhood of z=0 there exists a formal fundamental solution Y of (1) of the form \(Y=\hat{F}(z)z^{J}Ue^{Q}\) where \(\hat{F}(z)\) is a formal (usually divergent) meromorphic matrix, U,J are constant matrices and
$$Q=\mathrm{diag} \bigl(q_1 \bigl(z^{-\frac{1}{p}} \bigr),\ldots, q_n \bigl(z^{-\frac{1}{p}} \bigr) \bigr), \quad \mathrm{with}\ q_i \in X\mathbb{C}[X]\ \mathrm{and}\ p\geq1. $$
The Katz rank of the connection is the rational number
$$\kappa(\nabla)=\frac{1}{p}\max\deg_{1\leq i\leq n}q_i, $$
and is usually computed as the minimum Poincaré rank of the connection obtained after a suitable ramification z1/p of the variable. However, this is not needed in the tropical setting.

Theorem 5

Let\(\pi_{\varLambda}:\mathcal{M}_{\varLambda} \rightarrow L_{\varLambda}\)be the tropical nearest point projection map of the Gérard–Levelt membrane\(\mathcal{M}_{\varLambda}\)of any latticeΛonto its attached tropical linear space L. Then the Katz rankκ(∇) of the connectionsatisfies
$$\kappa(\nabla)=\min \bigl\{k\in\mathbb{R}^+\,|\,\pi_{\varLambda } \bigl(u^{n}_k \bigr)=\pi_{\varLambda} \bigl(u^{n-1}_k \bigr) \bigr\}\quad \mbox{\textit{for any lattice}}\ \varLambda. $$


The Katz rank is the minimum Poincaré rank of the connection ∇H induced on the pure algebraic extension H=K[T]/(TNz) of K with N=lcm(1,2,…,n) (see e.g. [18] or [6]). If we put ζ for the class of T, then \(\mathrm{Mat}((\nabla_{H})_{\zeta\frac{d}{d\zeta}},(e\otimes 1))=N\mathrm{Mat}(\nabla_{z\frac{d}{dz}},(e))\). Thus if X(z) satisfies \(z\frac{d}{dz}X(z)=A(z)X(z)\) the system satisfied by Y(ζ)=X(ζN) is
$$\zeta\frac{d}{d\zeta}Y(\zeta)=NA \bigl(\zeta^N \bigr)Y(\zeta). $$
Put \(\tilde{A}(\zeta)=NA(\zeta^{N})\). The sequence \((\tilde{A}_{k})_{k\in\mathbb{N}}\) defined by relation (14) of iterated \(\zeta\frac{d}{d\zeta}\)-derivatives of \(\tilde{A}\) satisfies
$$\tilde{A}_k(\zeta)=N^kA_k \bigl( \zeta^N \bigr). $$
Let q be the valuated matroid defined by \(q(\omega)=w(\det\tilde{M}(\zeta)_{\omega})\), for any n-subset ω of indices of the columns of \(\tilde{M}(\zeta)\) with respect to the ζ-adic valuation w. By construction we have The lattice \(N_{H}=\sum_{i=1}^{m} \mathcal{O}_{H} \zeta^{-u_{i}}v_{i}\otimes1\) has tropical representation in Lq as the projection of the point u∈ℤm with respect to the matroid q=Np. By corollary 10, we have \(m(\nabla_{H})=\min\{k\in\mathbb{N}\,|\,\pi_{N_{H}}(u^{n}_{k})=\pi_{N_{H}}(u^{n-1}_{k})\}\). On the other hand, \(\kappa(\nabla)=\frac{1}{N}m(\nabla_{H})\) holds. Therefore, we get
$$\kappa(\nabla)=\min \biggl\{k\in\frac{1}{N}\mathbb{N}\,\Big\vert\, \pi_{\varLambda} \bigl(u^{n}_k \bigr)=\pi_{\varLambda } \bigl(u^{n-1}_k \bigr) \biggr\}. $$
This formula holds for any extension H′ of degree divisible by the denominator s of κ(∇). Since the minimum is attained, the formula also holds in the limit, yielding the claimed result. □

Example 12

For Example 2, the saturation matrix M has size 4×20, and the lattice \(F^{4}_{k}(\mathcal{O}^{4})\) is therefore represented by the point
$$u^{4}_k=(0,0,0,0,-k,-k,-k,-k,-2k,\ldots,-3k,-4k,-4k,-4k,-4k) \in\mathbb{R}^{20}. $$
The matrix M is too long to be displayed entirely, so we show here only the principal parts of its entries. Accordingly the lattice \(F^{3}_{k}(\mathcal{O}^{4})\) is represented by
$$u^{3}_k=(0,0,0,0,-k,-k,-k,-k,-2k,\dots,-3k,-6,-5,-5,-6). $$
Similar computations as those in Example 11 give \(\pi_{\varLambda}(u^{4}_{k})=\pi_{\varLambda}(u^{3}_{k})\) if and only if \(k\geq\frac{3}{2}\), hence m=2 and \(\kappa=\frac{3}{2}\).

5 A projection algorithm on a tropical linear space

The Blue and Red rules from [16] recalled in Sect. 3 have unfortunately a high computational complexity, since they involve iterating over cardinality \(m\choose n\) sets. In our case, it is especially impractical since for the Gérard–Levelt membrane, we have mn2. In this section, we present an efficient algorithm, inspired by Ardila’s work on ordinary matroids [1], to compute the projection of a point x∈ℝm onto the tropical linear space Lp attached to a valuated matroid p.

5.1 Valuated matroids

Let us recall the setup of valuated matroids, and fix the notations that we will use. For the results listed in this section, we refer to [22], although their definition, following [8], comes with the opposite sign. Let E be a finite set, and a map p:2E→ℝ=ℝ∪{∞}. Let \(\mathcal{B}=\{ B\subset E \,|\, p(B)\neq\infty\}\). The pair (E,p) is a valuated matroid if \(\mathcal{B}\neq \emptyset\) and for \(B,B'\in\mathcal{B}\) and uBB′ there exists vB′∖B such that
$$ p(B)+p \bigl(B' \bigr)\geq p\bigl(B\cup\{v\}\backslash \{u\}\bigr)+p \bigl(B'\cup\{u\}\backslash\{v\} \bigr). $$
A subset \(B\in\mathcal{B}\) is called a base of p. In particular, \(\mathcal{B}\) is the set of bases of an ordinary matroid P on E that we call the matroid underlyingp. A base B is minimal if p(B)≤p(B′) for any base \(B'\in\mathcal{B}\). Any vector of the form
$$X(B,v)= \bigl(p\bigl(B\cup\{v\}\backslash\{u\}\bigr)-p(B),\: u\in E \bigr) $$
for some base B and vEB is by definition a circuit of p. If X is a circuit of p, its support
$$\overline{X}=\{e\in E \,|\,X_e\neq\infty\} $$
is a circuit of the matroid P. More precisely, it is the fundamental circuit of B and v, that is, the unique circuit of P included in B∪{v}. Similarly, any vector of the form
$$X^*(B,v)= \bigl(p\bigl(B\cup\{u\}\backslash\{v\}\bigr)-p(B), u\in E \bigr) $$
for some base B and vB is thus a cocircuit of p.
Some important features of circuits and cocircuits of p are in fact encoded in the underlying matroid P. For any circuit C of P, the set of circuits of p that have C as support is of the form
$$X+\alpha(1,\ldots,1)\quad \mathrm{for}\ \alpha\in\mathbb{R}. $$
Conversely, for any circuit X of p, X+α(1,…,1) for α∈ℝ is a circuit of p. The same result applies to cocircuits. Recall the following result.

Lemma 13

Any circuit (resp. cocircuit) ofPcontainingvEcan be represented as the fundamental circuit (resp. cocircuit) of a baseBsuch thatvB (resp. vB).


Let C be a circuit of P. By definition, for any vC, the set C∖{v} is contained in some base B. Therefore CB∪{v} holds. But there is a unique circuit satisfying this condition. Since the cocircuits are the circuits of the dual matroid, the same result holds. □

In what follows, we will speak by abuse of notation of the fundamental circuit or cocircuit of B and v for a valuated matroid p. This is harmless as long as the results that we state are invariant up to the addition of a constant. If we need to specify a representative, we will often use the only one with non-negative coordinates and with minimum coordinate equal to 0, or with some fixed value at some element of E.

For any x∈ℝm, the map
$$p_x(B)=p(B)-\sum_{b\in B}x_b $$
extended to all 2E by px(A)=∞ for \(A\notin\mathcal{B}\) defines also, as is well known, a valuated matroid on E.

Lemma 14

IfXis any circuit ofp, thenX+xis a circuit ofpx, and ifXis a cocircuit ofp, thenXxis a cocircuit ofpx.


By the definition of a circuit of p, circuits of px have coordinates Hence, Xx(B,v)=X(B,v)+xxv(1,…,1). Similarly, we have Hence, \(X^{*}_{x}(B,v)=X(B,v)-x+x_{v}(1,\ldots,1)\). By the projectivity property of circuits and cocircuits, the result is established. Since the sets of bases for p and px coincide, these are indeed the only circuits and cocircuits of px. □

5.2 The projection algorithm

A valuated matroid \(p:{E\choose{n}} \rightarrow\mathbb{R}_{\infty}\) of rank n over a finite set E=[m] induces a tropical linear space Lp defined by (11). This subspace of \(\mathbb{R}_{\infty}^{m}\) corresponds (up to sign) to what Dress and Terhalle call the tight span of a valuated matroid, except for the fact that, while Lp is invariant by translation by (1,…,1), the tight-span consists of only one point in every orbit (see [24]). In this section, we present an efficient algorithmic method to compute the tropical projection from ℝm onto Lp that generalises results obtained by Ardila for ordinary matroids in [1].

Proposition 15

Letpbe a valuated matroid of ranknon [m], and letuE. The following conditions are equivalent:
  1. (i)

    ubelongs to at least one minimal base ofp.

  2. (ii)

    uis never the unique minimum in a circuit ofp.

  3. (iii)

    uis minimal in some cocircuit ofp.



(i)⇒(iii): Assume that B is a minimal base containing u. Let C=X(B,u) be the fundamental cocircuit of B and u. By definition, we have
$$C^*_v=p\bigl(B\cup\{v\}\backslash\{u\}\bigr)-p(B) \geq0=C^*_u. $$
That is, u is minimal in the cocircuit of B and u.
(iii)⇒(ii): suppose that u is the unique minimum for p on a circuit C. Assume that C is a cocircuit of p where u is minimal. By assumption, we have
$$C_u<C_{u'}\quad \mathrm{and}\quad C^*_u\leq C^*_{u'}\quad \mathrm{for}\ u'\neq u. $$
Accordingly, C+C has a unique minimum at u. By orthogonality of circuits and cocircuits ([22], Theorem 3.11, p. 204), the set of indices that minimise C+C cannot have cardinality one. Therefore, the contradiction is established.
Let us finally prove (ii)⇒(i): consider a minimum base B. If uB, let C=X(B,u) be the circuit generated by B and u. By assumption, the minimum in C is attained at vu. The support of C is equal to the fundamental circuit of B and u for the ordinary matroid underlying p. Therefore, B∪{u}∖{v} is a base of p and
$$p\bigl(B\cup\{u\}\backslash\{v\}\bigr)-p(B)\leq p \bigl(B\cup\bigl \{u'\bigr\}\backslash \{v\} \bigr)-p(B)\quad \mathrm{for}\ u'\in C. $$
Putting v=u′ we get p(B∪{u}∖{v})≤p(B). Since we assumed that B was minimal, we get (i). □

Therefore we get the following characterisation of the (finite part of the) tropical linear space Lp.

Proposition 16

Letx∈ℝm, and letpbe a valuated matroid of ranknon [m]. The following are equivalent.
  1. (i)


  2. (ii)

    Every element ofEbelongs at least to onex-minimal base ofp.

  3. (iii)

    Every circuit ofpcontains at least twox-minimal elements.

  4. (iv)

    Every element ofEisx-minimal in at least one cocircuit ofp.



(i) and (iii) are equivalent by the definition of Lp (cf. [16]). The remaining assertions are obtained by applying Proposition 15 to the valuated matroid px. □

Note that the previous characterisation does not apply when x has an infinite coordinate, since px is then no longer a valuated matroid. However, xu=∞ happens only when u does not belong to any base. To deal with this case, one can either restrict to loop-free matroids (which means removing any 0 vectors in the membrane case), or put \(\pi_{L_{p}}(x)_{u}=\infty\).

The computation of \(\pi_{L_{p}}(x)\) can be performed independently for every coordinate of the vector x. For a given uE, there is a (unique) normalisation of a circuit C of p containing u such that \(C^{x}_{u}=x_{u}\).

Proposition 17

IfuEviolates any one of the three conditions of Proposition 15 for the valuated matroidpx, thenusatisfies them for the modified vector\(x'=(x_{1},\ldots,x'_{u},\ldots,x_{m})\)with
$$x'_u=\max_{C\ni u}\min_{e\in C\backslash\{u\}} C^x_e, $$
where all the circuits are normalised so that\(C^{x}_{u}=x_{u}\). Moreover, the conditions of Proposition 15 are not satisfied atufor\(x''=(x_{1},\ldots,x'_{u}-\varepsilon,\ldots,x_{m})\)withε>0.


By assumption, u is the unique x-minimum over some circuit \(\widetilde{C}\) containing u. The support of such a circuit C can be defined as \(\overline{C}=X(B,u)\) the fundamental circuit of u and a base \(B\not\ni u\). The x-value at \(e\in\overline{C}\) of the circuit C is of the form \(C^{x}_{e}=p(B\cup\{u\}\backslash\{e\})-p(B)+x_{e}+ \alpha\) for some constant α∈ℝ, so we may choose as representative of any circuit C containing u the only one such that \(C^{x}_{u}=x_{u}\), namely the one defined by \(C^{x}_{e}=p(B\cup\{u\}\backslash\{e\})-p(B)+x_{e}\).

Then by assumption
$$x'_u\geq\min_{e\in C\backslash\{u\}}\widetilde{C}^{x}_e> \widetilde{C}^{x}_u=x_u, $$
and for any circuit Cu, we have
$$C^{x'}_e=\left \{ \begin{array}{l@{\quad}l} C^{x'}_e & \mathrm{if}\ e\neq u\\[6pt] C^{x'}_u=x'_u &\mathrm{if}\ e=u. \end{array} \right . $$
$$C^{x'}_u\geq\min_{e\in C\backslash\{u\}}C^x_e= \min_{e\in C\backslash\{u\}}C^{x'}_e $$
so u cannot be the unique x′-minimum of any circuit containing u. On the other hand, there exists a circuit C containing u such that \(\min_{e\in C\backslash\{u\}}C^{x}_{e}=x'_{u}\). Putting \(x''_{u}=x'_{u}-\varepsilon\) for any ε>0, then u will be the x″-unique minimum over the circuit C. Thus x′ is the smallest vector that corrects the value of x at u. □

Theorem 6

Letx∈ℝmand letBbe anx-minimal base ofp. Then\(\omega=\pi_{L_{p}}(x)\)can be computed in the following way:
$$\omega_i=\left \{ \begin{array}{l@{\quad}l} x_i & \mathrm{if}\ i\in B\\[6pt] \min_{u\neq i}(p(B\cup\{i\}\backslash\{u\})-p(B)+x_u) & \mathrm{otherwise.} \end{array} \right . $$
Moreover, Bis alsoω-minimal.


If iB holds, then i is x-minimal in the fundamental cocircuit X(B,i). Therefore all conditions of Proposition 15 apply to i. Otherwise, let X(B,i) be the fundamental circuit of B and i, normalised so that X(B,i)i=xi. According to Proposition 17, we have to prove that \(\min_{e\neq i}(X(B,i)_{e}+x_{e})=\max_{C\ni i}\min_{e\in C\backslash\{ i\}} C^{x}_{e}\). By construction, ≤ holds. Moreover, it is sufficient to prove the result for x=0. So assume that B is minimum and X(B,i)i=0. We want to show that minei(X(B,i)e)≥mineC∖{i}Ce for any circuit C containing i.

Let C=X(B′,i) be any circuit containing i. Consider uC∖{i} such that mineiX(B,i)e=X(B,i)u. By construction, B1=B∪{i}∖{u} is a base. Since iB1B′, we apply axiom (17) and deduce that there exists jB′∖B1, such that Reordering terms, and inserting p(B), we get
$$p\bigl(B\cup\{i\}\backslash\{u\}\bigr)-p(B)+p(B)-p\bigl(B\cup\{j\}\backslash\{u \} \bigr)\geq p \bigl(B'\cup\{i\}\backslash\{j\} \bigr) -p \bigl(B' \bigr), $$
that is
$$X(B,i)_{u}+p(B)-p\bigl(B\cup\{j\}\backslash\{u\}\bigr)\geq X \bigl(B',i \bigr)_j=C_j. $$
Since B is a minimal base, we have p(B)−p(B∪{j}∖{u})≤0, hence
$$X(B,i)_{u}\geq\min_{e\in C\backslash\{i\}}C_e\quad \mathrm{since}\ j \in C\backslash\{i\}, $$
and the result is established.
For the last statement, consider a px-minimum base B. Let B′ be a base which is adjacent to B, that is, such that there exist ij in E with B∖{i}=B′∖{j}. Let Δ=pω(B′)−pω(B). Then we have Hence B is locally ω-minimum. According to [9], the global minimum of pω is obtained by choosing the local minimum at each step. Hence B is also a global minimum of pω. □
Theorem 6 yields an efficient method to compute the tropical projection \(\pi_{L_{p}}(x)\), described in Algorithm 1. Instead of considering all circuits or cocircuits as in the Red and Blue rules, our algorithm uses only the fundamental circuits of a minimal base.
Algorithm 1

Tropical projection on a linear tropical space Lp

Computing a minimal base B of px can be performed by the greedy Algorithm 2, originally described in [9].
Algorithm 2

Greedy algorithm for a minimal base

5.3 The greedy algorithm for a realisable valuated matroid

Let VKn, and consider a family of mn vectors M=(v1,…,vm) of rank n. For any basis (e) of V, consider the valuated matroid
$$p_{(e)}(\omega)=v \bigl(\det_{(e)}(M_{\omega}) \bigr) \quad \mathrm{where}\ M_{\omega}=(v_{\omega_1},\ldots,v_{\omega_n}). $$

Lemma 18

A baseBis minimal forp(e)if and only if it is minimal forp(ε)for any other basis (ε) ofV.


For any family F of n vectors, one has det(ε)(F)=det(e)(F)det(ε)(e). Therefore
$$p_{(\varepsilon)}(\omega) = v \bigl(\det_{(\varepsilon)}(M_{\omega }) \bigr) = v \bigl(\det_{(e)}(M_{\omega}) \bigr)+v \bigl( \det_{(\varepsilon)}(e) \bigr) = p_{(e)}(\omega)+v \bigl( \det_{(\varepsilon)}(e) \bigr). $$
Hence p(ε)p(e) is a constant map, and the claim is established. □
It follows that in the minimising step of the greedy Algorithm 2, it does not matter in which basis we compute the determinant. This simplifies the search for the minimising vector. Let us assume that the first n vectors form a base B, in which we express all other vectors. Namely, we consider the matrix
$$ \mathbf{M}_B=\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c}I_n & C_{n+1} & \ldots& C_m \end{array} \right )\quad \mathrm{where}\ C_i\ \mbox{are the coordinates of}\ v_i\ \mathrm{in}\ B. $$
Moreover we consider B to be the starting point of the greedy algorithm, except that we will reorder successively the vectors of B in another order than 1,2,…,n. Indeed, consider matrices of the form
$$A_{i,u}=\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & & u_1 & & 0 \\ & \ddots&\vdots& &\\ \vdots& & u_i & &\vdots\\ & & \vdots& \ddots& \\ 0 & & u_n & & 1 \end{array} \right ). $$
It is clear that detAi,u=ui and that
$$(A_{i,u})^{-1}=A_{i,\tilde{u}}\quad \mathrm{holds}\quad \tilde{u}= \biggl(-\frac{u_1}{u_i},\ldots,\frac{1}{u_i},\ldots,-\frac {u_n}{u_i} \biggr). $$
Therefore, if (i1,j1) are indices such that \(v(\mathbf{M}_{i_{1},j_{1}})\leq v(\mathbf{M}_{i,j})\), then replacing the vector \(e_{i_{1}}\) in the base B by the vector \(v_{j_{1}}\) will indeed attain the local minimum. The next step consists of
  1. 1.

    Define \((e_{1},\ldots,u_{i_{1}},\ldots,e_{n})\) as new base B′.

  2. 2.

    Create MB=(Ai,u)−1MB.

Then one starts the same procedure again with the matrix MB: this time the minimum should be taken over rows ii1 and columns jj1. This is explained in Algorithm 3.
Algorithm 3

Greedy algorithm for a realisable matroid


  1. 1.

    Which has also been independently obtained by Rincón (personal communication).



Supported in part by DFG grant MO 1048/6-1.

I have benefited from very interesting and stimulating discussions with Federico Ardila, Josephine Yu, Michael Joswig, Annette Werner, Felipe Rincón and Stéphane Gaubert, held at the Tropical Geometry Workshop at the CIEM in Castro Urdiales (Spain) in December 2011.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Ardila, F.: Subdominant matroid ultrametrics. Ann. Comb. 8, 379–389 (2004) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Babbitt, D.G., Varadarajan, V.S.: Formal reduction theory of meromorphic differential equations: a group theoretic view. Pac. J. Math. 109, 1–80 (1983) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Balser, W., Jurkat, W.B., Lutz, D.A.: A general theory of invariants for meromorphic differential equations. Funkc. Ekvacioj 22, 197–221 (1979) MathSciNetMATHGoogle Scholar
  4. 4.
    Barkatou, M.A.: A rational version of Moser’s algorithm. In: Proceedings ISSAC, Montréal, Canada, pp. 297–302. ACM, New York (1995) Google Scholar
  5. 5.
    Cohen, G., Gaubert, S., Quadrat, J.P.: Duality and separation theorems in idempotent semimodules. Linear Algebra Appl. 379, 395–422 (2004) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Corel, E.: On Fuchs’ relation for linear differential systems. Compos. Math. 140, 1367–1398 (2004) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Deligne, P.: Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, vol. 163. Springer, Berlin (1970) MATHGoogle Scholar
  8. 8.
    Dress, A.W.M., Terhalle, W.: The tree of life and other affine buildings. In: Doc. Math., Extra Volume ICM 1998, Part III, pp. 565–574 Google Scholar
  9. 9.
    Dress, A.W.M., Wenzel, W.: Valuated matroid: a new look at the greedy algorithm. Appl. Math. Lett. 3, 33–35 (1990) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Fuchs, L.I.: Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coeffizienten. J. Reine Angew. Math. 66, 121–160 (1866) MATHCrossRefGoogle Scholar
  11. 11.
    Gaubert, S., Katz, R.: Minimal half-spaces and external representation of tropical polyhedra. J. Algebr. Comb. 33(3), 325–348 (2011) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gérard, R., Levelt, A.H.M.: Invariants mesurant l’irrégularité en un point singulier d’un système d’équations différentielles linéaires. Ann. Inst. Fourier 23(1), 157–195 (1973) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Jurkat, W.: Meromorphe Differentialgleichungen. Lecture Notes in Mathematics, vol. 637. Springer, Berlin (1978) MATHGoogle Scholar
  14. 14.
    Katz, N.: Nilpotent connections and the monodromy theorem. Applications of a result of Turrittin. Publ. Math. 39, 176–232 (1970) Google Scholar
  15. 15.
    Hilali, A., Wazner, A.: Formes super-irréductibles des systèmes différentiels linéaires. Numer. Math. 50(4), 429–449 (1987) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Joswig, M., Sturmfels, B., Yu, J.: Affine buildings and tropical geometry. Albanian J. Math. 4, 187–211 (2007) MathSciNetGoogle Scholar
  17. 17.
    Keel, S., Tevelev, J.: Geometry of Chow quotiens of Grassmannians. Duke Math. J. 134(2), 259–311 (2006) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Levelt, A.H.M.: Jordan decomposition for a class of singular differential operators. Ark. Mat. 13, 1–27 (1975) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Levelt, A.H.M.: Stabilizing differential operators. A method for computing invariants at irregular singularities. In: Differential Equations and Computer Algebra, pp. 181–228. Academic Press, London (1991) Google Scholar
  20. 20.
    Manin, Y.: Moduli fuchsiani. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 19(1), 113–126 (1965). Serie III MathSciNetMATHGoogle Scholar
  21. 21.
    Moser, J.: The order of a singularity in Fuchs’ theory. Math. Z. 72, 379–398 (1960) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Murota, K., Tamura, A.: On circuit valuation of matroids. Adv. Appl. Math. 26(3), 192–225 (2001) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Rincón, F.: Computing linear tropical spaces. Preprint, arXiv:1109.4130
  24. 24.
    Speyer, D.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Werner, A.: A tropical view on Bruhat–Tits buildings and their compactifications. Cent. Eur. J. Math. 9(2), 390–402 (2011) MathSciNetMATHCrossRefGoogle Scholar

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© The Author(s) 2012

Authors and Affiliations

  1. 1.Georg-August-Universität GöttingenGöttingenGermany

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