Adv. Geom. 4 (2004), 389–411 Advances in Geometry ( de Gruyter 2004 The tropical Grassmannian David Speyer and Bernd Sturmfels* (Communicated by G. M. Ziegler) Abstract. In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Gro¨bner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plu¨cker relations. It parametrizes all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian G2; n equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians o¤er a natural generalization of the space of trees. Their faces correspond to monomial-free initial ideals of the Plu¨cker ideal. The tropical Grassmannian G3; 6 is a simplicial complex glued from 1035 tetrahedra. 1 Introduction The tropical semiring ðR U fyg; min; þÞ is the set of real numbers augmented by in- finity with the tropical addition, which is taking the minimum of two numbers, and the tropical multiplication which is the ordinary addition [10]. These operations sat- isfy the familiar axioms of arithmetic, e.g. distributivity, with y and 0 being the a1 an Ptwo neutral elements. Tropical monomials x1 ...xn represent ordinary linear forms i aixi, and tropical polynomials X a1 a2 an H Nn A R Fðx1; x2; ...; xnÞ¼ Cax1 x2 ...xn ; with A ; Ca ; ð1Þ a A A n represent piecewise-linear convex functions FP: R ! R. To compute FðxÞ, we take n A the minimum of the a‰ne-linear forms Ca þ i¼1 aixi for a A. We define the trop- ical hypersurface TðFÞ as the set of all points x in Rn for which this minimum is at- tained at least twice, as a runs over A. Equivalently, TðFÞ is the set of all points x A Rn at which F is not di¤erentiable. Thus a tropical hypersurface is an ðn À 1Þ- dimensional polyhedral complex in Rn. The rationale behind this definition will become clear in Section 2, which gives *Partially supported by the National Science Foundation (DMS-0200729). 390 David Speyer and Bernd Sturmfels a self-contained development of the basic theory of tropical varieties. For further background and pictures see [14, §9]. Every tropical variety is a finite intersection of tropical hypersurfaces (Corollary 2.3). But not every intersection of tropical hyper- surfaces is a tropical variety (Proposition 6.3). Tropical varieties are also known as logarithmic limit sets [2], Bieri–Groves sets [4], or non-archimedean amoebas [7]. Trop- ical curves are the key ingredient in Mikhalkin’s formula [9] for planar Gromov– Witten invariants. The object of study in this paper is the tropical Grassmannian Gd; n which is a poly- n hedral fan in RðÞd defined by the ideal of quadratic Plu¨cker relations. All of our main results regarding Gd; n are stated in Section 3. The proofs appear in the subsequent sections. In Section 4 we prove Theorem 3.4 which identifies G2; n with the space of 20 phylogenetic trees in [5]. A detailed study of the fan G3; 6 H R is presented in Sec- tion 5. In Section 6 we introduce tropical linear spaces and we prove that they are parametrized by the tropical Grassmannian (Theorem 3.6). In Section 7 we show that the tropical Grassmannian G3; 7 depends on the characteristic of the ground field. 2 The tropical variety of a polynomial ideal Let K be an algebraically closed field with a valuation into the reals, denoted deg : K à ! R. We assume that 1 lies in the image of deg and we fix t A K à with degðtÞ¼1. The corresponding local ring and its maximal ideal are RK ¼fc A K : degðcÞ d 0g and MK ¼fc A K : degðcÞ > 0g: The residue field k ¼ RK =MK is algebraically closed. Given any ideal I H K½x¼K½x1; x2; ...; xn; we consider its a‰ne variety in the n-dimensional algebraic torus over K, VðIÞ¼fu A ðK ÃÞn : f ðuÞ¼0 for all f A Ig: Here K à ¼ Knf0g. In all our examples, K is the algebraic closure of the rational function field CðtÞ and ‘‘deg’’ is the standard valuation at the origin. Then k ¼ C, and if c A C½t then degðcÞ is the order of vanishing of c at 0. Every polynomial in K½x maps to a tropical polynomial as follows. If X a1 an A à A f ðx1; ...; xnÞ¼ cax1 ...xn with ca K for a A: ð2Þ a A A and Ca ¼ degðcaÞ, then tropð f Þ denotes the tropical polynomial F in (1). The following definitions are a variation on Gro¨bner basis theory [13]. Fix A Rn a1 an w . The w-weight of a term ca Á x1 ...xn in (2) is degðcaÞþa1w1 þÁÁÁþanwn. ~ The initial form inwð f Þ of a polynomial f is defined as follows. Set f ðx1; ...; xnÞ¼ w1 wn Àn ~ f ðt x1; ...; t xnÞ. Let n be the smallest weight of any term of f , so that t f is a The tropical Grassmannian 391 Àn ~ non-zero element in RK ½x. Define inwð f Þ as the image of t f in k½x. We set inwð0Þ¼0. For K ¼ CðtÞ and k ¼ C this means that the initial form inwð f Þ is a polynomial in C½x. Given any ideal I H K½x its initial ideal is defined to be inwðIÞ¼hinwð f Þ : f A Ii H k½x: Theorem 2.1. For an ideal I H K½x the following subsets of Rn coincide: (a) The closure of the set fðdegðu1Þ; ...; degðunÞÞ : ðu1; ...; unÞ A VðIÞg; (b) The intersection of the tropical hypersurfaces Tðtropð f ÞÞ where f A I; n (c) The set of all vectors w A R such that inwðIÞ contains no monomial. The set defined by the three conditions in Theorem 2.1 is denoted TðIÞ and is called the tropical variety of the ideal I. Variants of this theorem already appeared in [14, Theorem 9.17] and in [7, Theorem 6.1], without and with proof respectively. Here we present a short proof which is self-contained. Proof. First we show that (b) contains (a). As (b) is clearly closed, it is enough to consider any point w ¼ðdegðu1Þ; ...; degðunÞÞ in the set (a) and show it lies in (b). For any f A I we have f ðu1; ...; unÞ¼0 and this implies that the minimum in the definition of F ¼ tropð f Þ is attained at least twice at w. This condition is equivalent to inwð f Þ not being a monomial. This shows that (a) is contained in (b), and (b) is contained in (c). It remains to prove that (c) is contained in (a). Consider any vector à n w in (c) such that w ¼ðdegðv1Þ; ...; degðvnÞÞ for some v A ðK Þ . Since the image of the valuation is dense in R and the set defined in (a) is closed, it su‰ces to prove that w ¼ðdegðu1Þ; ...; degðunÞÞ for some u A VðIÞ. By making the change of coordinates À1 xi ¼ xi Á vi , we may assume that w ¼ð0; 0; ...; 0Þ. Since inwðIÞ contains no monomial and since k is algebraically closed, by the Null- à n stellensatz there exists a point u A VðinwðIÞÞ H ðk Þ . Let m denote the maximal ideal in k½x corresponding to u. Let S be the set of polynomials f in RK ½x whose reduc- tion modulo MK is not in m. Then S is a multiplicative set in RK ½x disjoint from I. Consider the induced map À1 À1 j : RK ! S RK ½x=S ðI V RK ½xÞ: We claim that j is injective. Suppose not, and pick c A RK nf0g with jðcÞ¼0, so we can find f A S such that cf A I. Since cÀ1 exists in K, this implies f A I which is a contradiction. The injectivity of j implies that there is some minimal prime P of the ring on the n right hand side such that P RK K is a proper ideal in K½x=I. There exists a maximal n ideal of K½x=I containing P RK K, and since K is algebraically closed, this maximal à n ideal has the form hx1 À u1; ...; xn À uni for some u A VðIÞ H ðK Þ . We claim that ui A RK and ui G ui mod MK . This will imply degðu1Þ¼ÁÁÁ¼degðunÞ¼0 and hence complete the proof. Consider any xi À ui A I. By clearing denominators, we get aix À bi A I V RK ½x 392 David Speyer and Bernd Sturmfels with bi=ai ¼ ui, and not both ai and bi lie in MK .Ifai A MK , then aix À bi G Àbi mod MK . Hence inwðIÞ contains the reduction of bi modulo MK , which is a à unit of K and hence equals the unit ideal. This is a contradiction. If ai B MK and Àbi=ai Z ui mod MK then the reduction of aix À bi modulo MK does not lie in m. À1 This means that aix À bi A S and is a unit of S RK ½x,soP is the unit ideal. But then P is not prime, also a contradiction. This completes the proof. r The key point in the previous proof can be summarized as follows: Corollary 2.2.