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Gérard-Levelt membranes Eduardo Corel∗ 2 Abstract 1 We present an unexpected application of tropical convexity to the determi- 0 2 nation of invariants for linear systems of differential equations. We show that the classical Gérard-Levelt lattice saturation procedure can be geometrically n understood in terms of a projection on the tropical linear space attached to a a subset of the local affine Bruhat-Tits building, that we call the Gérard-Levelt J membrane. This provides a way to compute the true Poincaré rank, but also 1 the Katz rank of a meromorphic connection without having to perform gauge 1 transformsnorramificationsofthevariable. Wefinallypresentanefficientalgo- ] rithmtocomputethistropicalprojectionmap,generalisingArdila’smethodfor A thecaseoftheBergmanfantothecaseofthetight-spanofavaluatedmatroid. C h. Contents t a m 1 Meromorphic connections 3 1.1 Gérard-Levelt’ssaturated lattices . . . . . . . . . . . . . . . . . . . . . 3 [ 1 2 Tropical convexity and lattices 5 v 1 3 The Gérard-Levelt membranes 6 4 3.1 Tropical computation of the Katz rank . . . . . . . . . . . . . . . . . . 7 4 2 4 A projection algorithm on a tropical linear space 8 . 1 4.1 Valuated matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 0 4.2 An algorithm for the projection on the tight-span . . . . . . . . . . . . 9 2 1 : v Introduction i X Given a meromorphic linear differential system on the Riemann sphere, r a dX =A(z)X with A(z)∈M (C(z)), (1) n dz it is important to determine whether a singularity of A is a regular singular point for the system (1). Unlike with scalar linear differential equations, for which there is a purelyalgebraicconditionontheordersofthepolesofthecoefficientsduetoL.Fuchs [Fu], a system (1) can display arbitrarily high pole orders at a regular singularity. Consider the differential system, expanded in the neighbourhood of the singular point (assumed for simplicity to be z =0) as follows, where we put θ =z d , dz 1 θX = A ziX with p>0 and A 6=0 if p>0. (2) zp i 0 Xi>0 ∗E-mail: [email protected] 1 2 TheintegerpistraditionallyknownasthePoincaré rankp(A)ofthesystem. Finding the type of singularityinvolvesknowingthe minimum value m(A), sometimes known as the true Poincaré rank, of this rank under gauge transformations A =P−1AP −P−1θP with P = P zk ∈GL (K) where K =C((z)). (3) [P] k n kX>k0 Severallinesofresearchhavebeenopenedtotacklethisproblem. Themostclassi- caltriestoiterativelyconstructasuitablegaugetransformationP,usuallycoefficient by coefficient in the series expansion. Featured methods rely on the linear algebra over C involved by equation (3), like Moser and continuators ([Mo, H-W]), whose methods are widely used nowadaysin computer algebra, or other researcherssuch as [Ju, Ba-J-L], while [B-V] use Lie group theoretic tools. The nature of a singularity of A can also considered from point of view of meromorphic connections [De], and especially, as a question of stability of certain lattices under the differential operator induced by the connection [M, Ka]. We focus here specifically on the approach of saturating lattices used by Gérard and Levelt [G-L]: the true Poincaré rank m(∇) is the minimum integer k such that the sequence of k-saturated lattices (recalled in section 1.1) eventually stabilises. Recent work has shown close relations between the geometric framework of the Bruhat-TitsbuildingofSL(K),forsomediscretevaluedfieldsuchasK =C((z)),and tropicalconvexity[K-T,J-S-Y,W]. WeshowherethatGérardandLevelt’sapproach can be formulated and efficiently computed in this framework,as a projectionon the tropical linear space L attached to the valuated matroid p corresponding to a given p membrane in the Bruhat-Tits building. ResultsofKeelandTevelev[K-T]showthat,whenlatticesareinasamemembrane of the Bruhat-Tits building, they are homothetic if and only if they are projected on the same point of a tropical linear space by an explicit nearest point projection map ([J-S-Y], see also [C-G-Q, G-K]). We need to be able to tell whether lattices are actuallyequal. Weshowthattheprojectionmaprespectsthevaluation,andprojects only equal lattices on a same point (theorem 2). We then construct the Gérard-Levelt membrane that contains all the k-saturated lattices (proposition 1), and give a tropical version of Gérard-Levelt’s lattice stabili- sationcriterion(corollary1). Third,weshowthatthesameformulaamazinglyallows for the computation of a more subtle invariant, namely the Katz rank of the connection (theorem 3), without having to either to compute a single gauge transformation nor perform the usually required ramification of the variable. Theprojectionmapgivenin[J-S-Y]hasunfortunatelyahighcomplexity. Wegive insection4anefficientalgorithmtocomputethisprojectionmapontothelineartrop- ical space attached to a valuated matroid. We generalise the algorithmic approach to tropical projection developed by Ardila [Ar] for ordinary matroids, to the case of valuatedmatroidsdefinedbyDressandcollaborators[D-W]. Thealgorithmpresented here, which computes the nearest ℓ -projection on the tight-span of a valuated ma- ∞ troid, has a wider applicability than the differential computations explained in the previous parts, especially in phylogenetics [D-T]. Acknowledgements I have benefited from very interesting and stimulating discussions with Federico Ardila, Josephine Yu, Michael Joswig, Annette Werner, Felipe Rincoń and Stéphane 3 Gaubert, held at the Tropical Geometry Workshop at the CIEM in Castro Urdiales (Spain) in December 2011. 1 Meromorphic connections A meromorphic connection is a map ∇:V ≃Kn −→Ω(V)=V ⊗ Ω1(K) which is K C C-linear and satisfies the Leibniz rule ∇(fv)=v⊗df +f∇v for f ∈K and v ∈V. The matrix Mat(∇,(e)) is given by ∇e = − n e ⊗Ω for a basis (e). A basis j i=1 i ij change P ∈GLn(K) gauge-transforms the matPrix of ∇ by Ω =P−1ΩP −P−1dP. (4) [P] Contracting with zk+1 d yields a differential operator ∇ , and system (1) is the dz k expression of ∇ (v)=0 in the basis (e). −1 A lattice Λ in V is a free sub-O-module of rank n, that is a module of the form n Λ= Oe for some basis (e) of V. i Mi=1 The Poincaré rank of ∇ on the lattice Λ is defined as the integer p (∇)=−v (Λ+∇ (Λ))=min{k∈N|∇ (Λ)⊂Λ}=max(−v(Ω )−1,0). (5) Λ Λ 0 k ij i,j The true Poincaré rank m(∇) = min p (∇) characterises the nature of the sin- Λ Λ gularity of (1), in the sense that z =0 is regular if and only if m(∇)=0. 1.1 Gérard-Levelt’s saturated lattices For any vector e∈V and any derivation τ ∈Der(K/C), define for ℓ∈N the family Zℓ(e)=(e,∇ e,...,∇ℓe). τ τ τ The module Oℓ(e) induced over O by Zℓ(e) only depends on the valuation v(τ) of τ τ the derivation τ. We can therefore restrict ourselves to the particular derivations τ =zk+1 d for k ∈N. Inthis case,we put ∇ =∇ , andnote Zℓ(e) andOℓ(e) the k dz τk k k k corresponding objects. For k >1, Gérardand Levelt define the lattices Fℓ(Λ)=Λ+∇ Λ+···+∇ℓΛ. k k k Note that n Fℓ(Λ)= Zℓ(e ) for any basis (e ,...,e ) of Λ. k k i 1 n Xi=1 Theorem 1 (Gérard, Levelt). The true Poincaré rank m(∇) of ∇ is m(∇)=min{k∈N|Fn(Λ)=Fn−1(Λ)} for any lattice Λ⊂V. k k 4 This means that k > m(∇) if and only if ∇ (Fn−1(Λ)) ⊂ Fn−1(Λ) for some k k k (equivalently, any) lattice Λ in V, that is, that the Poincaré rank on Fn−1(Λ) is at k most k. Stated otherwise,finding the true Poincarérank is finding the largestlattice whose Poincarérank is bounded by its index in the following sequence Fn−1(Λ)⊃···⊃Fn−1(Λ)⊃Λ. (6) 0 p−1 Letusextendthis notationto multi-indices. Letℓ>0,andletα=(α ,...,α )∈ 1 ℓ Zℓ be an integer multi-index of length |α| = ℓ and weight w(α) = α +···+α . Let 1 ℓ us define also the partial multi-indices α =(α ,...,α ) and |j 1 j ∇α =∇ ◦···◦∇ . αℓ α1 Let by convention α =ǫ and ∇ǫ =id for the empty sequence ǫ. Let finally Oα(e) |0 V be the O-module spanned by the sequence Zα(e)=(∇α|je)06j6|α|. Lemma 1. For any α=(α ,...,α )∈Zℓ, one has 1 ℓ ∇α =zw(α)P (∇ ) where P (X)=X(X+α )···(X +w(α ))∈Z[X]. α 0 α 1 |ℓ−1 Proof. 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 P =1, so assume that ǫ there exists P ∈Z[X] such that ∇α =zw(α)P (D) for |α| 6ℓ. Let β ∈Zℓ+1. Then α α by definition, we have ∇β = zβℓ+1D◦∇β|ℓ = zβℓ+1D◦(zw(β|ℓ)Pβ|ℓ(D)) = zβℓ+1 w(β|ℓ)zw(β|ℓ)Pβ|ℓ(D)+zw(β|ℓ)D◦Pβ|ℓ(D) (cid:16) (cid:17) = zw(β) w(β )P (D)+P (D)◦D |ℓ β|ℓ β|ℓ = zw(β)(cid:0)P (D)◦(D+w(β )) . (cid:1) β|ℓ |ℓ (cid:0) (cid:1) Indeed, P (D) commutes with D since it has by assumption constant coefficients. β|ℓ The result follows, since we have then P (X)=P (X)(X +w(β )). β β|ℓ |ℓ Lemma 2. Let Λ be a lattice in V. For any ℓ∈N and α∈Nℓ, the O-module Oα(e) is spanned over O by the family e,zα1∇0e,...,zw(α|ℓ−1)∇ℓ0−1e . (cid:16) (cid:17) Proof. According to lemma 1, the family Zℓ−1(e) is relatedto Zℓ−1(e) by the matrix α 0 P =AzWα where W =diag(0,α ,...,w(α )), α 1 |ℓ−1 andAisanuppertriangularintegermatrixwithdiagonalentriesequalto1,therefore A ∈ SLℓ(Z) ⊂ GLℓ(O). The families zWαZ0ℓ−1(e) and Zαℓ−1(e) are related by the matrix P˜ =z−WαP =z−WαAzWα whose entries are Aijzw(α|j)−w(α|i). Since A is upper triangular,and the partial sums α +···+α are non-negative, the i j matrix P˜ is in GL (O), and therefore both families span the same O-module. n 5 2 Tropical convexity and lattices Let M = {d ,...,d } be lines in V such that d +···+d = V, and consider the 1 m 1 m subset of Λ defined by [M]={ℓ +···+ℓ |ℓ is a lattice in d }. 1 m i i Following Keel and Tevelev, who call in [K-T] the set induced by [M] modulo homo- thety a membrane, we call this the affine membrane spanned by M. For a choice A = (v ,...,v ) of non-zero vectors in the lines d , any lattice in 1 m i the membranedefinedby M ={d ,...,d }canbe represented(nonuniquely)byan 1 m integer valued point as follows: a lattice point u∈Zm corresponds to the lattice m Λu = Oz−uivi. (7) Xi=1 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, where the operations are ∞ x⊕y =min(x,y) and x⊙y =x+y for x,y ∈R . ∞ An affine membrane M and a basis (e) of V determine a valuated matroid p:[m]n −→ R ∞ ω 7−→ v(det M ) (e) ω where M = (v ,...,v ) is the subfamily of vectors of M indexed by ω. To a ω ω1 ωn valuated matroid p of rank n over [m] there is a tropical linear space L ⊂ Rm p ∞ attached as follows [m] L = x∈Rm |∀τ ∈ , min p(τ\{τ })+x is attained twice . (8) p (cid:26) ∞ (cid:18)n+1(cid:19) 16i6n+1 i τi (cid:27) Depending on the authors, L is said to be a tropical convex cone ([C-G-Q]) or a p convex polytope ([J-S-Y]) in Rm. Both definitions mean that ∞ λ⊙u⊕µ⊙v ∈L for any λ,µ∈R and u,v ∈L . p ∞ p According to [C-G-Q, G-K, J-S-Y], for x∈Rm, the formula ∞ π (x)=min{w ∈L |w>x} where the minimum is taken coordinate-wise, (9) Lp p defines the nearest point projection map π : Rm −→ L . There are at least two Lp ∞ p other known ways to characterise or compute π (x). Lp Blue Rule. Adapting[Ar],theauthorsof[J-S-Y]showthatπ (x)=(w ,...,w ) Lp 1 m with w = min max(p(σ∪{i})−p(σ∪{j})+x ). i j σ∈([m]) j6=σ n−1 6 Red Rule. Similarly, starting with v = (0,...,0) ∈ Rm, for every τ ∈ [m] such n+1 thatα=min16i6n+1p(τ\{τi})+xτi isonlyattainedonce,sayatτi,comput(cid:0)eγ =(cid:1)β−α where β is the second smallest number in that collection, and put v :=max(v ,γ). τi τi Then π (x)=x+v. Lp Theorem 2 (Keel-Tevelev). The nearest point projection map π : Rm −→ L Lp ∞ p induces a bijection Ψ between [M] and the lattice points in L M p Ψ (Λ )=π (u ,...,u ). M u Lp 1 m Inparticular,Mu =(z−u1v1,...,z−umvm)andMu′ =(z−u′1v1,...,z−u′mvm)span the same lattice Λ if and only if π (u)=π (u′). Lp Lp Proof. Let Λ= mi=1Oz−uivi, and let w=πLp(u). According to [K-T], th. 4.17(see also [J-S-Y] th. P18), there exists α ∈R such that wi = vΛ(vi)+α for all 1 6i 6 m. By definition, vΛ(x) = max{k ∈ Z|x ∈ zkΛ}. Accordingly, we have z−wivi ∈ z−αΛ, and thus α>0. By formula (9), we get α=0 and thus π (u)=(v (v ),...,v (v )). (10) Lp Λ 1 Λ m By construction, if u′ > u, then z−uivi = z(u′i−ui)z−u′ivi ∈ Oz−u′ivi, for 1 6 i 6 m, hence Λ ⊂ Λ . Since in particular w > u holds, we get Λ ⊂ Λ . Conversely, we u′ w have Λw = mi=1Oz−wivi ⊂ Λ. Therefore, if πLp(u) = πLp(u′) then Λu = Λu′. The converse follPows directly from (10). 3 The Gérard-Levelt membranes Proposition 1. Fix a basis (e) of Λ, and ℓ>0. Let [M ] be the membrane spanned ℓ by the vectors (∇je ) . Then Fℓ′(Λ)∈[M ] for all k >0 and ℓ′ 6ℓ. 0 i 16i6n,06j6ℓ k ℓ Proof. For the considered basis (e), the lattice L=Fℓ′(Λ) satisfies k L=Oα(e )+···+Oα(e ) with α=(0,k,...,kℓ′). 1 n Reordering terms as (e ,...,e ,∇ e ,...∇ e ,...,∇ℓ′e ), formula (7) and lemma 2 1 n 0 1 0 n 0 n imply that L can be represented in the membrane [M ] by the lattice point ℓ′ (0,...,0,−k,...,−k,...,−kℓ′,...,−kℓ′). ntimes ntimes ntimes | {z } | {z } | {z } Since by definition, z−vΛ(v)v ∈ Λ holds for any v ∈ V, the module L can also be represented as an element of the membrane [M ] by ℓ (0,...,0,−k,...,−k,...,−kℓ′,...,−kℓ′,v (∇ℓ′+1e ),...,v (∇ℓe )). Λ 0 1 Λ 0 n ntimes ntimes ntimes | {z } | {z } | {z } The lattices Fℓ(Λ) for 06ℓ6n cantherefore allbe seen as elements of the same k membrane [M ]. n Definition 1. M =[M ] is called the Gérard-Leveltmembrane attached to Λ. For Λ n any basis (e), the lattice Fℓ(Λ) is represented by the lattice point k uℓ =(0,...,0,−k,...,−k,...,−kℓ,...,−kℓ,v (∇ℓ+1e ),...,v (∇ne )). k Λ 0 1 Λ 0 n ntimes ntimes ntimes | {z } | {z } | {z } 7 If Mat(∇ ,(e)) = A for an basis (e) of Λ, then M is described in (e) by the 0 Λ n×n(n+1) matrix d M= I A ··· A where A =(z +A)A and A =I . (11) n n k+1 k 0 n dz (cid:0) (cid:1) The tropical projection π onto the tropical linear space L attached to the Λ Λ Gérard-Levelt membrane M maps a point u to a unique representative. Check- Λ ingifk >m(∇)requirestoknowifthelatticepointsun−1 andun representthe same k k lattice, that is π (un)=π (un−1). Λ k Λ k Corollary 1. For any Λ, we have m(∇)=min{k ∈N|π (un)=π (un−1)}. Λ k Λ k 3.1 Tropical computation of the Katz rank The tropical setting is compatible with the ramification of the variable. This implies the following result. Theorem 3. Let π :M −→L be the tropical nearest point projection map of the Λ Λ Λ Gérard-Levelt membrane M of any lattice Λ onto its attached tropical linear space Λ L. Then the Katz rank κ(∇) of the connection ∇ satisfies κ(∇)=min{k∈R+|π (un)=π (un−1)} for any lattice Λ. Λ k Λ k Proof. The Katz rank is the minimum Poincaré rank of the connection ∇ induced H on the pure algebraic extension H =K[T]/(TN −z) of K with N =lcm(1,2,...,n) (see e.g. [Cor]). If we put ζ for the class of T, then Mat((∇ ) ,(e ⊗ 1)) = H ζ d dζ NMat(∇ ,(e)). Thus if X(z) satisfies z d X(z) = A(z)X(z) the system satisfied z d dz dz by Y(ζ)=X(ζN) is d ζ Y(ζ)=NA(ζN)Y(ζ). dζ Put A˜(ζ) = NA(ζN). The sequence (A˜k)k∈N defined by relation (11) of iterated ζ d -derivatives of A˜ satisfies dζ A˜ (ζ)=NkA (ζN). k k Letq be the valuatedmatroiddefinedby q(ω)=w(detM˜(ζ) ),foranyn-subsetω of ω indicesofthecolumnsofM˜(ζ)withrespecttotheζ-adicvaluationw. Byconstruction we have q(ω)=w(detM˜(ζ) ) = w(detM(ζN) ) ω ω = w((detM(ζ) )N) ω = Nw(detM(ζ) ) ω = Np(ω). The lattice NH = mi=1OHζ−uivi ⊗1 has tropical representation in Lq as the projection of the pointPu ∈ Zm with respect to the matroid q = Np. By corollary 1, we have m(∇ ) = min{k ∈ N|π (un) = π (un−1)}. On the other hand, H NH k NH k κ(∇)= 1m(∇ ) holds. Therefore, we get N H 1 κ(∇)=min{k∈ N|π (un)=π (un−1)}. N Λ k Λ k This formula holds for any extension H′ of degree divisible by the denominator s of κ(∇), hence the result also holds in the limit, yielding the claimed result. 8 Example (Pflu¨gel-Barkatou). Let dX/dz =AX with −5z−2 5z−1 −2z−1 −9z−2  5z−3 3z−2 2z−2 −4z−2 A= . 4z−1 −5z−1 −5z−2 2    2−2z −5z−1 3z−2 −6z−2  z3  un =(0,...,−3k,−4k−4k−4k−4k) k un−1 =(0,...,−3k,−6,−5,−5,−6). k 3 3 One gets π(un)=π(un−1) ⇐⇒ k > , therefore m(∇)=2 but actually κ(∇)= . k k 2 2 4 A projection algorithm on a tropical linear space The Blue and Red rules from [J-S-Y] recalled in section 3.1 have unfortunately a high computational complexity, since it involves loops over cardinality m sets. In n our case, it is especially impractical since for the Gérard-Levelt membra(cid:0)ne(cid:1), we have m ∼ n2. In this section, we present an efficient algorithm, inspired by Ardila’s work on ordinary matroids [Ar], to compute the projection of a point x ∈ Rm onto the tropical linear space L attached to a valuated matroid p. p 4.1 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 [Mu-Ta], although their definition, following [D-T], comes with the opposite sign. Let E be a finite set, and a map p : 2E −→ R = R∪{∞}. Let B = {B ⊂ E|p(B) 6= ∞}. The pair (E,p) is a ∞ valuated matroid if B 6= ∅ and for B,B′ ∈ B and u ∈ B\B′ there exists v ∈ B′\B such that p(B)+p(B′)>p(B∪{v}\{u})+p(B′∪{u}\{v}). A subset B ∈ B is called a basis of p. In particular, B is the set of bases of an ordinary matroid P on E, that we call the matroid underlying p. Any vector of the form X(B,v)=(p(B∪{v}\{u})−p(B), u∈E) for some basis B and v ∈E\B is a circuit of p. If X is a circuit of p, its support X ={e∈E|X 6=∞} e isacircuitofthe matroidP. Moreprecisely,itisthe fundamentalcircuitofB andv, that is the unique circuit of P included in B∪{v}. Similarly, any vector of the form X∗(B,v)=(p(B∪{u}\{v})−p(B), u∈E) for some basis B and v ∈B 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 +α(1,...,1) for α∈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. 9 Lemma 3. Any circuit (resp. cocircuit) of P containing v ∈E can be represented as the fundamental circuit (resp. cocircuit) of a basis B such that v ∈/ B (resp. v ∈B). Proof. LetCbeacircuitofP. Bydefinition,foranyv ∈C,thesetC\{v}iscontained insomebasisB. ThereforeC ⊂B∪{v}holds. Butthereisauniquecircuitsatisfying thiscondition. Sincethecocircuitsarethecircuitsofthedualmatroid,thesameresult holds. Inwhatfollows,wewillspeakbyabuseofnotationofthefundamental(co-)circuit 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,wewilloftenusethe onlyonewithnon-negativecoordinatesandwith minimum coordinate equal to 0, or with some fixed value at some element of E. For any x∈Rm, the map p (B)=p(B)− x x b bX∈B extended to all 2E by p (A)=∞ for A∈/ B defines another valuated matroid on E. x Lemma 4. If X is any circuit of p, then X +x is a circuit of p , and if X∗ is a x cocircuit of p, then X∗−x is a cocircuit of p . x Proof. By the definition of a circuit of p, circuits of p have coordinates x X (B,v) = p (B∪{v}\{u})−p (B) for some B 6∋v x u x x = p(∪{v}\{u})− x −p(B)+ x b b b∈B∪X{v}\{u} bX∈B = X(B,v) −x +x . u v u Hence, X (B,v)=X(B,v)+x−x (1,...,1). Similarly, we have x v X∗(B,v) = p (B∪{u}\{v})−p (B) x u x x = p(∪{u}\{v})− x −p(B)+ x b b b∈B∪X{u}\{v} bX∈B = X∗(B,v) +x −x . u v u Hence,X∗(B,v)=X(B,v)−x+x (1,...,1). Bytheprojectivitypropertyofcircuits x v and cocircuits, the result is established. Since the sets of bases for p and p coincide, x these are indeed the only (co)circuits of p . x 4.2 An algorithm for the projection on the tight-span A valuated matroid p : E −→ R of rank n over a finite set E = [m] induces a n ∞ tropicallinearspaceLp d(cid:0)efi(cid:1)nedby(8). ThissubspaceofRm∞ corresponds(uptosign) to what Dress and Terhalle call the tight span of a valuated matroid. In this section, we present an efficient algorithmic method to compute the tropical projection from Rm ontoL ,thatgeneralisesresultsobtainedbyArdilaforordinarymatroidsin[Ar]. p Proposition 2. Let p be a valuated matroid of rank n on [m], and let u ∈ E. The following conditions are equivalent. i) u belongs to at least one minimal basis of p. 10 ii) u is never the unique minimum in a circuit of p. iii) u is minimal in some cocircuit of p. Proof. i)⇒iii): Assume that B is a minimalbasis containingu. LetC∗ =X∗(B,u) be the fundamental cocircuit of B and u. By definition, we have C∗ =p(B∪{v}\{u})−p(B)>0=C∗. v u That is, u is minimal in the cocircuit of B and u. iii) ⇒ ii): suppose that u in 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 <C and C∗ 6C∗ for u′ 6=u. u u′ u u′ Accordingly, C +C∗ has a unique minimum at u. By orthogonality of circuits and cocircuits([Mu-Ta], th. 3.11,p. 204),the setofindices that minimise C+C∗ cannot have cardinality one. Therefore, the contradiction is established. Let us finally prove ii) ⇒ i): consider a minimum basis B. If u ∈/ B, let C = X(B,u) be the circuit generated by B and u. By assumption, the minimum in C is attained at v 6= u. 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 basis of p and p(B∪{u}\{v})−p(B)6p(B∪{u′}\{v})−p(B) for u′ ∈C. Puttingv =u′ wegetp(B∪{u}\{v})6p(B). SinceweassumedthatB wasminimal, we get i). Therefore we get the following characterisationof the (finite part of the) tropical linear space L . p Proposition 3. Let x∈Rm, and let p be a valuated matroid of rank n on [m]. The following are equivalent. i) x∈L . p ii) Every element of E belongs at least to one x-minimal basis of p. iii) Every circuit of p contains at least 2 x-minimal elements. iv) Every element of E is x-minimal in at least one cocircuit of p. Proof. i) and iii) are equivalent by the definition of L (cf. [J-S-Y]). The remaining p assertions are obtained by applying proposition 2 to the valuated matroid p . x Note that the previous characterisation does not apply when x has an infinite coordinate,since p is then nolongeravaluatedmatroid. However,x =∞happens x u only when u does not belong to any basis. The computation of π (x) can be performed independently for every element of Lp the vector x. For a given u∈E, there is a (unique) normalisation of a circuit C of p containing u such that Cx =x . u u Proposition 4. If u∈E violates any one of the three conditions of proposition 2 for the matroid p , then u satisfies them for the modified vector x′ =(x ,...,x′,...,x ) x 1 u m with x′ =max min Cx, u e u∈C e∈C\{u} where all the circuits are normalised so that Cx = x . Moreover, the conditions of u u proposition 2 are not satisfied at u for x′′ =(x ,...,x′ −ε,...,x ) with ε>0. 1 u m