Gérard–Levelt membranes
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DOI: 10.1007/s1080101203878
 Cite this article as:
 Corel, E. J Algebr Comb (2013) 37: 757. doi:10.1007/s1080101203878
Abstract
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 tightspan of a valuated matroid.
Keywords
Meromorphic connections Tropical convexity Valuated matroids1 Introduction
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 ksaturated 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 discretevalued 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 socalled membrane) can be faithfully represented as the set of integervalued points of the tropical linear space defined by a tropical Plücker vector (or valuated matroid). If a membrane \(\mathcal{M}\) is generated by vectors v _{1},…,v _{ m }, a lattice Λ in \(\mathcal{M}\) admits as nonunique representative vector any u∈ℤ^{ m } 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]).
Theorem 1
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
(Pflügel–Barkatou)
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 L _{ p } is the tropical linear space [24] attached to it, then we have the following result.
Theorem 2
The algorithm based on this result,^{1} which computes the nearest ℓ _{∞}projection on the tightspan of a valuated matroid, has a wider applicability than the differential computations explained in the previous parts, especially in phylogenetics [8].
2 Meromorphic connections
2.1 Gérard–Levelt’s saturated lattices
Theorem 3
(Gérard, Levelt)
Lemma 3
Proof
Lemma 4
Proof
3 Tropical convexity and lattices
Membranes spanned by m lines in the Bruhat–Tits building have a faithful representation as tropical linear spaces in mdimensional space.
Example 5
 Blue Rule:

\(\pi_{L_{p}}(x)=(w_{1},\ldots,w_{m})\) with$$w_i=\min_{\sigma\in{{[m]}\choose{n1}}}\max_{j\neq \sigma} \bigl(p\bigl(\sigma\cup \{i\}\bigr)p\bigl( \sigma\cup\{j\}\bigr)+x_j \bigr). $$
 Red Rule:

Starting with v=(0,…,0)∈ℝ^{ m }, 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 τ _{ i }, compute γ=β−α 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
If we let \(z^{u}v=(z^{u_{1}}v_{1},\ldots,z^{u_{m}}v_{m})\), then z ^{−u } v and z ^{−u′} v span the same lattice Λ if and only if \(\pi_{L_{p}}(u)=\pi_{L_{p}}(u^{\prime})\).
Proof
As mentioned in [16], as soon as the projection \(\pi_{L_{p}}(u)\) is computed, one can also determine a basis of the lattice Λ _{ u }.
Lemma 6
Let M={v _{1},…,v _{ m }} be a set of vectors of rank n in V, and let L _{ p } be the associated tropical linear space. Let \((w_{1},\ldots,w_{m})=\pi_{L_{p}}(u)\). For any nsubset τ⊂{1,…,m} such that \(p(\tau)w_{\tau_{1}}\cdotsw_{\tau_{n}}\) is minimal, the subfamily \((v_{\tau_{1}},\allowbreak \ldots,v_{\tau_{n}})\) is an \(\mathcal{O}\)basis of Λ _{ u }.
Proof
As a consequence, one can find a basis of the lattice Λ _{ u } 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
4 The Gérard–Levelt membranes
Proposition 8
Fix a basis (e) of Λ, and ℓ≥0. Let [M _{ ℓ }] 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 all k≥0 and ℓ′≤ℓ.
Proof
The lattices \(F^{\ell}_{k}(\varLambda)\) for 0≤ℓ≤n can therefore all be seen as elements of the same membrane [M _{ n }].
Definition 9
Corollary 10
For any Λ, we have \(m(\nabla)=\min\{k\in\mathbb{N}\,\,\pi_{\varLambda }(u^{n}_{k})=\pi_{\varLambda}(u^{n1}_{k})\}\).
Example 11
4.1 Tropical computation of the Katz rank
Theorem 5
Proof
Example 12
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 m∼n ^{2}. 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∈ℝ^{ m } onto the tropical linear space L _{ p } attached to a valuated matroid p.
5.1 Valuated matroids
Lemma 13
Any circuit (resp. cocircuit) of P containing v∈E can be represented as the fundamental circuit (resp. cocircuit) of a base B such that v∉B (resp. v∈B).
Proof
Let C be a circuit of P. By definition, for any v∈C, the set C∖{v} is contained in some base B. Therefore C⊂B∪{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 nonnegative coordinates and with minimum coordinate equal to 0, or with some fixed value at some element of E.
Lemma 14
If X is any circuit of p, then X+x is a circuit of p _{ x }, and if X ^{∗} is a cocircuit of p, then X ^{∗}−x is a cocircuit of p _{ x }.
Proof
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 L _{ p } 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 L _{ p } is invariant by translation by (1,…,1), the tightspan 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 ℝ^{ m } onto L _{ p } that generalises results obtained by Ardila for ordinary matroids in [1].
Proposition 15
 (i)
u belongs to at least one minimal base of p.
 (ii)
u is never the unique minimum in a circuit of p.
 (iii)
u is minimal in some cocircuit of p.
Proof
Therefore we get the following characterisation of the (finite part of the) tropical linear space L _{ p }.
Proposition 16
 (i)
x∈L _{ p }.
 (ii)
Every element of E belongs at least to one xminimal base of p.
 (iii)
Every circuit of p contains at least two xminimal elements.
 (iv)
Every element of E is xminimal in at least one cocircuit of p.
Proof
(i) and (iii) are equivalent by the definition of L _{ p } (cf. [16]). The remaining assertions are obtained by applying Proposition 15 to the valuated matroid p _{ x }. □
Note that the previous characterisation does not apply when x has an infinite coordinate, since p _{ x } is then no longer a valuated matroid. However, x _{ u }=∞ happens only when u does not belong to any base. To deal with this case, one can either restrict to loopfree 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 u∈E, there is a (unique) normalisation of a circuit C of p containing u such that \(C^{x}_{u}=x_{u}\).
Proposition 17
Proof
By assumption, u is the unique xminimum 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 xvalue 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 \(C^{x}_{e}=p(B\cup\{u\}\backslash\{e\})p(B)+x_{e}\).
Theorem 6
Proof
If i∈B holds, then i is xminimal 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 }=x _{ i }. 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 min_{ e≠i }(X(B,i)_{ e })≥min_{ e∈C∖{i}} C _{ e } for any circuit C containing i.
5.3 The greedy algorithm for a realisable valuated matroid
Lemma 18
A base B is minimal for p _{(e)} if and only if it is minimal for p _{(ε)} for any other basis (ε) of V.
Proof
 1.
Define \((e_{1},\ldots,u_{i_{1}},\ldots,e_{n})\) as new base B′.
 2.
Create M _{ B′}=(A _{ i,u })^{−1} M _{ B }.
Acknowledgements
Supported in part by DFG grant MO 1048/61.
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
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