[proofplan] The proof uses the circuit-minimum definition of the Bergman fan and the residue-field initial-ideal criterion for tropical membership. It specializes this criterion to a linear space whose circuit equations have coefficients of valuation zero. A circuit of the matroid gives a minimal linear dependence among the coordinate functionals, so evaluating that dependence at a torus point forces the minimum of the coordinate valuations on the circuit to occur at least twice. Conversely, if a weight vector satisfies the circuit minimum condition, then every initial circuit form has at least two terms, and the initial linear ideal has no monomial; the fundamental theorem of tropical geometry for linear ideals then lifts the weight vector to a valued point after a valued [field extension](/page/Field%20Extension). The closure in the definition of $\operatorname{Trop}(L)$ accounts for possible limiting valuation vectors. [/proofplan]

[step:Write the defining linear equations using matroid circuits] For each $e \in E$, define the coordinate projection $\pi_e: K^E \to K$ by $\pi_e((x_d)_{d\in E})=x_e$, and let $\ell_e := \pi_e|_L: L \to K$ be its restriction to $L$. Also let $\ell_{0,e}:=\pi_e|_{L_0}:L_0\to K_0$ denote the corresponding coordinate restriction on $L_0$. The scalar-extension identification $L=L_0\otimes_{K_0}K$ identifies the $K$-linear span of $(\ell_e)_{e\in E}$ with the scalar extension of the $K_0$-linear span of $(\ell_{0,e})_{e\in E}$. Equivalently, for every subset $A\subset E$, let $\Phi_A:K^A\to \operatorname{Hom}_K(L,K)$ be the $K$-[linear map](/page/Linear%20Map) defined by $\Phi_A((b_e)_{e\in A})=\sum_{e\in A} b_e\ell_e$, and let $\Phi_{0,A}:K_0^A\to \operatorname{Hom}_{K_0}(L_0,K_0)$ be the $K_0$-linear map defined by $\Phi_{0,A}((b_e)_{e\in A})=\sum_{e\in A} b_e\ell_{0,e}$. Then $\ker \Phi_A=(\ker \Phi_{0,A})\otimes_{K_0}K$. Thus a minimal $K$-linear dependence among the coordinate restrictions on $L$ comes from a minimal $K_0$-linear dependence among the coordinate restrictions on $L_0$. Let $C \subset E$ be a circuit of $M(L)$. By definition of a circuit, the family $(\ell_e)_{e \in C}$ is minimally linearly dependent over $K$. Hence there are coefficients $a_e \in K_0^\times$, unique up to common scalar in $K_0^\times$, such that \begin{align*} \sum_{e \in C} a_e \ell_e = 0 \end{align*} as a linear functional on $L$. Since the valuation is zero on $K_0^\times$, we have \begin{align*} v(a_e)=0 \end{align*} for every $e \in C$. Equivalently, the circuit $C$ determines a linear equation on $L$: \begin{align*} \sum_{e \in C} a_e x_e = 0 \end{align*} for every point $x=(x_e)_{e \in E} \in L$, with all $a_e \in K_0^\times$. [/step]

[step:Use each circuit equation to force the Bergman minimum condition]By the definition of the affine Bergman fan fixed in the theorem statement, a vector $w\in\mathbb{R}^E$ lies in $\widetilde{\mathcal B}(M(L))$ if and only if, for every circuit $C$ of $M(L)$, the minimum of $(w_e)_{e\in C}$ is attained at least twice. Let $K'/K$ be a valued field extension with valuation $v':K'\to \mathbb{R}\cup\{\infty\}$, and let $x=(x_e)_{e \in E} \in (L\otimes_K K') \cap (K'^\times)^E$. Define the valuation vector $w=(w_e)_{e \in E} \in \mathbb{R}^E$ by \begin{align*} w_e := v'(x_e) \end{align*} for each $e \in E$. Fix a circuit $C$ of $M(L)$, and choose the circuit equation \begin{align*} \sum_{e \in C} a_e x_e = 0 \end{align*} with $a_e \in K_0^\times$. Since $K_0 \subset K \subset K'$ and the valuation is zero on $K_0^\times$, we have \begin{align*} v'(a_e x_e)=v'(a_e)+v'(x_e)=w_e \end{align*} for every $e \in C$. We claim that $\min_{e \in C} w_e$ is attained at least twice. If it were attained at a unique element $e_0 \in C$, then the nonarchimedean valuation inequality applied to the equality \begin{align*} a_{e_0}x_{e_0} = -\sum_{e \in C \setminus \{e_0\}} a_e x_e \end{align*} would give \begin{align*} w_{e_0} = v'(a_{e_0}x_{e_0}) = v'\left(-\sum_{e \in C \setminus \{e_0\}} a_e x_e\right) \geq \min_{e \in C \setminus \{e_0\}} v'(a_e x_e) = \min_{e \in C \setminus \{e_0\}} w_e. \end{align*} This contradicts the strict inequality $w_{e_0}<w_e$ for every $e \in C \setminus \{e_0\}$. Therefore every valuation vector of a torus point of $L_{K'}$ satisfies the circuit condition. Since $\widetilde{\mathcal B}(M(L))$ is closed in $\mathbb{R}^E$, taking the Euclidean closure in the definition of $\operatorname{Trop}(L)$ gives \begin{align*} \operatorname{Trop}(L) \subset \widetilde{\mathcal B}(M(L)). \end{align*}[/step]

[guided]We prove the first containment by looking at one circuit at a time. A circuit $C$ gives one minimal linear relation among the coordinate functionals on $L$, and because the realization is defined over $K_0$, that relation can be written with coefficients in $K_0^\times$: \begin{align*} \sum_{e \in C} a_e x_e = 0. \end{align*} The hypothesis on the valuation is exactly what makes this relation unshifted tropically: for every $e \in C$, \begin{align*} v'(a_e x_e)=v'(a_e)+v'(x_e)=0+w_e=w_e. \end{align*} Now suppose the minimum of the numbers $(w_e)_{e \in C}$ were attained only once, say at $e_0$. Then $a_{e_0}x_{e_0}$ would be the unique summand of smallest valuation in the circuit equation. Moving it to the other side gives \begin{align*} a_{e_0}x_{e_0} = -\sum_{e \in C \setminus \{e_0\}} a_e x_e. \end{align*} The nonarchimedean valuation inequality says that the valuation of a finite sum is at least the minimum of the valuations of its summands. Hence \begin{align*} v'(a_{e_0}x_{e_0}) \geq \min_{e \in C \setminus \{e_0\}} v'(a_e x_e). \end{align*} Substituting $v'(a_e x_e)=w_e$ gives \begin{align*} w_{e_0} \geq \min_{e \in C \setminus \{e_0\}} w_e. \end{align*} This is impossible if $w_{e_0}$ is the unique strict minimum on $C$. Therefore the minimum on every circuit is attained at least twice. This proves that every actual valuation vector of a torus point of $L$ over any valued extension lies in the Bergman fan. The Bergman fan is defined by finitely many closed conditions of the form “the minimum on $C$ is attained at least twice,” so it is closed in $\mathbb{R}^E$. Therefore the closure of all such valuation vectors is still contained in $\widetilde{\mathcal B}(M(L))$.[/guided]

[step:Translate the Bergman condition into monomial-free initial circuit forms]Let $S_K := K[x_e : e \in E]$ be the [polynomial ring](/page/Polynomial%20Ring) with one variable for each element of $E$, and let $I(L) \subset S_K$ be the homogeneous linear ideal of polynomials vanishing on $L \subset K^E$. For a weight vector $w=(w_e)_{e \in E}\in\mathbb{R}^E$, choose a valued extension $F/K$ whose value group contains the subgroup of $\mathbb R$ generated by $v(K^\times)$ and the coordinates $w_e$, and let $v_F:F\to \mathbb R\cup\{\infty\}$ and $\kappa_F$ denote its valuation and residue field. For a nonzero polynomial $f=\sum_u c_u x^u\in F[x_e:e\in E]$, where the sum is over a finite set of exponent vectors $u\in\mathbb N^E$, define the weight-minimum map \begin{align*} m_w: F[x_e:e\in E]\setminus\{0\}\to\mathbb R \end{align*} by \begin{align*} m_w(f):=\min_u(v_F(c_u)+w\cdot u). \end{align*} After choosing $\lambda\in F^\times$ with $v_F(\lambda)=-m_w(f)$, define the residue initial-form map up to nonzero scalar by \begin{align*} \operatorname{in}_w: F[x_e:e\in E]\setminus\{0\}\to \kappa_F[x_e:e\in E]/\kappa_F^\times \end{align*} where $\operatorname{in}_w(f)$ is represented by the reduction of the sum of the terms $\lambda c_u x^u$ for which $v_F(c_u)+w\cdot u=m_w(f)$. Replacing $\lambda$ by another element of the same valuation multiplies this representative by a nonzero scalar in $\kappa_F$, so the ideal generated by all such initial forms is independent of that rescaling. If one passes to a further valued extension, the residue-field initial ideal is obtained by scalar extension, so the property of containing a monomial is unchanged. For a circuit $C$ with equation $f_C := \sum_{e \in C} a_e x_e \in I(L)$, where $a_e \in K_0^\times$, the $w$-initial form of $f_C$ is supported exactly on the set \begin{align*} C_{\min}(w) := \{e \in C : w_e = \min_{c \in C} w_c\}. \end{align*} Indeed, the coefficient valuations are all zero, so the terms of smallest weighted valuation in $f_C$ are precisely those indexed by $C_{\min}(w)$; reducing the coefficients cannot delete any of these terms because each $a_e$ has residue nonzero. Assume now that $w \in \widetilde{\mathcal B}(M(L))$. Then $|C_{\min}(w)| \geq 2$ for every circuit $C$, so no initial circuit form $\operatorname{in}_w(f_C)$ is a monomial. For a linear form $g=\sum_{e\in E} b_e x_e$, define its support by $\operatorname{supp}(g):=\{e\in E:b_e\neq 0\}$. Let $S_{K,1}$ denote the degree-one homogeneous part of $S_K$, and let $I(L)_1\subset S_{K,1}$ denote the $K$-[vector space](/page/Vector%20Space) of degree-one forms in $I(L)$. Define the degree-one residue initial space \begin{align*} W_w:=\operatorname{span}_{\kappa_F}\{\operatorname{in}_w(g):g\in I(L)_1\setminus\{0\}\}\subset \kappa_F[x_e:e\in E]_1. \end{align*} Here each $\operatorname{in}_w(g)$ is represented by any residue initial form of the nonzero linear form $g$, and the span is independent of the scalar choices. [claim:Initial monomials in a linear ideal are detected by circuit forms] Let $\mathcal C$ denote the set of circuits of $M(L)$. For every weight vector $w \in \mathbb{R}^E$, the valued initial ideal $\operatorname{in}_w(I(L))$ contains a monomial if and only if $\operatorname{in}_w(f_C)$ is a monomial for some circuit $C \in \mathcal C$. [/claim] [proof] The space $W_w$ was defined above as the residue-field span of all initial forms of nonzero elements of $I(L)_1$. We first justify that $\operatorname{in}_w(I(L))$ is the linear ideal generated by $W_w$. Since $I(L)$ is generated by $I(L)_1$, every $f\in I(L)$ can be written as a finite sum $f=\sum_j q_j g_j$ with $g_j\in I(L)_1$ and $q_j\in K[x_e:e\in E]$. Taking residue initial forms gives $\operatorname{in}_w(f)$ in the ideal generated by the initial degree-one space after collecting all summands of minimal total weighted valuation; possible cancellation only replaces the displayed minimal linear combination by the initial form of another element of $I(L)_1$. Thus all initial forms of elements of $I(L)$ lie in $(W_w)$, while the reverse inclusion holds because each element of $W_w$ is an initial form of an element of $I(L)$. Hence \begin{align*} \operatorname{in}_w(I(L))=(W_w). \end{align*} A linear ideal generated by a vector subspace contains a monomial if and only if it contains a variable. Indeed, after a residue-field linear change of coordinates, the quotient by this linear ideal is a polynomial ring, hence no nonzero monomial is zero in the quotient unless one of its variable factors is zero in the quotient. Translating back, this means some variable belongs to the linear ideal, and because the ideal is generated in degree one, that variable lies in $W_w$. A linear ideal generated by a vector subspace contains a monomial if and only if it contains a variable. Indeed, the quotient of a polynomial ring by a linear ideal is a polynomial ring over the residue field, hence an integral domain. If the class of a coordinate monomial is zero in this quotient, then the product of the classes of its coordinate factors is zero, so at least one coordinate variable has zero class and belongs to the linear ideal. Since the ideal is generated by the degree-one space $W_w$, that variable lies in $W_w$. We now use the standard initial-circuit theorem for a represented valuated matroid. In the form needed here, let $D\subset K^E$ be a finite-dimensional vector space of linear dependences of a represented matroid, give each coordinate $e$ the real weight $w_e$, and let $\operatorname{in}_w(D)$ be the residue-field span of the initial forms of nonzero vectors of $D$. Then the nonzero inclusion-minimal supports in $\operatorname{in}_w(D)$ are exactly the circuits of the initial matroid; equivalently, they are the inclusion-minimal nonempty sets among $\operatorname{supp}(\operatorname{in}_w(d_C))$, where $C$ ranges over the circuits of the original represented matroid and $d_C\in D$ is any circuit vector supported on $C$. This is the represented case of valuated-circuit elimination: the circuit vectors of a representation satisfy valuated circuit elimination after weighting the coordinates, and their initial supports are precisely the circuits of the initial matroid. We apply this theorem with $D=I(L)_1$, the dependence space of the coordinate functionals $(\ell_e)_{e\in E}$. The circuits of the original represented matroid are exactly the minimal supports of nonzero elements of $I(L)_1$, which are the circuits of $M(L)$. For such a circuit $C$, the earlier scalar-extension argument chooses a circuit vector $f_C=\sum_{e\in C}a_ex_e$ with every $a_e\in K_0^\times$, so all coefficient valuations are zero and $\operatorname{supp}(\operatorname{in}_w(f_C))=C_{\min}(w)$. Therefore the theorem identifies the nonzero inclusion-minimal supports in $W_w=\operatorname{in}_w(I(L)_1)$ with the minimal nonempty supports among these initial circuit forms. If $\operatorname{in}_w(I(L))$ contains a monomial, then the preceding paragraph gives a variable $x_{e_0}\in W_w$. The singleton $\{e_0\}$ is then an inclusion-minimal nonzero support in $W_w$. By valuated-circuit elimination, there is a circuit $C$ of $M(L)$ such that $\operatorname{supp}(\operatorname{in}_w(f_C))=\{e_0\}$, so $\operatorname{in}_w(f_C)$ is a monomial. Conversely, if $\operatorname{in}_w(f_C)$ is a monomial for some circuit $C$, then it belongs to $\operatorname{in}_w(I(L))$, so the initial ideal contains a monomial. [/proof] By the claim, a monomial appears in $\operatorname{in}_w(I(L))$ if and only if the initial form of some circuit equation is a monomial. Since no circuit initial form is a monomial, $\operatorname{in}_w(I(L))$ contains no monomial.[/step]

[guided]We now convert the Bergman circuit condition into the initial-ideal condition required for tropical lifting. Let $S_K := K[x_e : e \in E]$ be the polynomial ring and let $I(L)\subset S_K$ be the homogeneous linear ideal of all polynomials vanishing on $L$. For a weight vector $w=(w_e)_{e\in E}\in\mathbb{R}^E$, we use the valued initial ideal: terms are compared by $v(c_u)+w\cdot u$, the minimal terms are rescaled to valuation zero, and their coefficients are reduced in the residue field. Take a circuit $C$ of $M(L)$. Its circuit equation has the form $f_C := \sum_{e\in C} a_e x_e$, with every $a_e\in K_0^\times$. Since the valuation is zero on $K_0^\times$, the weighted valuation of the term $a_e x_e$ is exactly $w_e$. Hence the initial form $\operatorname{in}_w(f_C)$ is supported precisely on \begin{align*} C_{\min}(w):=\{e\in C: w_e=\min_{c\in C}w_c\}. \end{align*} No coefficient in this support disappears after residue reduction, because each $a_e$ is a unit of valuation zero. Since $w\in\widetilde{\mathcal B}(M(L))$, the minimum on each circuit is attained at least twice. Therefore $C_{\min}(w)$ has at least two elements, and $\operatorname{in}_w(f_C)$ is not a monomial. It remains to justify why checking only circuits checks the whole initial ideal. The ideal $I(L)$ is generated by its degree-one part, namely the vector space of all linear dependences among the coordinate restrictions $(\ell_e)_{e\in E}$. Let $S_{K,1}$ denote the degree-one homogeneous part of $S_K$, so $I(L)_1\subset S_{K,1}$. The degree-one residue initial space is \begin{align*} W_w:=\operatorname{span}_{\kappa_F}\{\operatorname{in}_w(g):g\in I(L)_1\setminus\{0\}\}\subset \kappa_F[x_e:e\in E]_1. \end{align*} Because every element of $I(L)$ is a polynomial combination of elements of $I(L)_1$, every initial form of an element of $I(L)$ lies in the ideal generated by $W_w$ after collecting the summands of minimal weighted valuation; if those minimal summands cancel, the remaining leading linear combination is the initial form of another element of $I(L)_1$. Conversely, every generator of $W_w$ is an initial form of an element of $I(L)$. Hence $\operatorname{in}_w(I(L))=(W_w)$. A linear ideal contains a monomial only if it contains a variable: the quotient by the linear ideal is a polynomial ring, hence an integral domain; if a coordinate monomial has zero class in the quotient, then one coordinate variable factor has zero class and therefore lies in the ideal. The remaining point is the circuit detection theorem for represented valuated matroids, and here is the exact form being used. For a finite represented matroid whose dependence space is $D\subset K^E$, and for coordinate weights $w_e$, the residue-field initial space $\operatorname{in}_w(D)$ has nonzero inclusion-minimal supports equal to the circuits of the initial matroid. Equivalently, those supports are the inclusion-minimal nonempty supports among the initial circuit vectors $\operatorname{in}_w(d_C)$, where $C$ runs over the original circuits and $d_C$ is a circuit vector supported on $C$. This is the represented valuated-circuit elimination theorem: weighted circuit vectors eliminate to give exactly the circuits of the initial represented matroid. We apply it with $D=I(L)_1$. This is legitimate because $I(L)_1$ is precisely the vector space of linear dependences among the represented family $(\ell_e)_{e\in E}$, and the original circuits of that representation are the circuits of $M(L)$. For each circuit $C$, the circuit vector is the equation $f_C=\sum_{e\in C}a_ex_e$ with $a_e\in K_0^\times$. Since all these coefficients have valuation zero, the support of $\operatorname{in}_w(f_C)$ is exactly $C_{\min}(w)$. Thus the nonzero inclusion-minimal supports in $W_w=\operatorname{in}_w(I(L)_1)$ are detected by the initial circuit forms used above. Therefore, if $\operatorname{in}_w(I(L))$ contained a monomial, then $W_w$ would contain some variable $x_{e_0}$. The singleton $\{e_0\}$ would be a minimal nonzero support in $W_w$, so the circuit detection theorem would produce a circuit $C$ with $\operatorname{supp}(\operatorname{in}_w(f_C))=\{e_0\}$. That would make $\operatorname{in}_w(f_C)$ a monomial. But we have shown that no circuit form has monomial initial form for a Bergman weight $w$. Hence $\operatorname{in}_w(I(L))$ contains no monomial.[/guided]

[step:Lift monomial-free initial degenerations to valued torus points]For an ideal $I\subset K[x_e:e\in E]$, define its affine zero set over a field extension $F/K$ by \begin{align*} V(I)(F):=\{x\in F^E: f(x)=0\text{ for every }f\in I\}. \end{align*} For each valued field extension $F/K$, define the scalar extension shorthand \begin{align*} L_F := L \otimes_K F \subset F^E, \end{align*} where the inclusion is induced by extending the coordinate embedding $L \subset K^E$ coefficientwise from $K$ to $F$. We now use the initial-ideal form of the [Fundamental Theorem of Tropical Geometry](https://doi.org/10.1090/gsm/161) for valued fields. The version needed here is the following exact-lifting statement. Let $I \subset K[x_e : e \in E]$ be an ideal, and let $w\in\mathbb R^E$. Compute residue-field initial ideals after passing, if necessary, to a valued extension whose value group contains the subgroup generated by $v(K^\times)$ and the coordinates $w_e$. Then $\operatorname{in}_w(I)$ contains no monomial if and only if there is a further algebraically closed valued field extension $K'/K$, whose value group may be taken to be a subgroup of $\mathbb R$ containing those values, and a point \begin{align*} x=(x_e)_{e \in E} \in V(I)(K') \cap (K'^\times)^E \end{align*} such that $v'(x_e)=w_e$ for every $e \in E$. The real-valued assertion follows by adjoining only the finitely generated ordered subgroup of $\mathbb R$ generated by $v(K^\times)$ and the coordinates of $w$, and then taking an [algebraic closure](/page/Algebraic%20Closure) with the extended valuation. We verify the hypotheses for $I=I(L)$. The ideal $I(L)$ is a homogeneous linear ideal in the polynomial ring $K[x_e : e \in E]$, the vector $w$ lies in $\mathbb{R}^E$, and the previous step proved that $\operatorname{in}_w(I(L))$ contains no monomial. The extension permitted by the theorem may be chosen with real-valued value group because only the [real numbers](/page/Real%20Numbers) $w_e$ and the original real value group are adjoined. Since $V(I(L))(K')$ is exactly the scalar extension $L_{K'}$, the lifting theorem gives a point \begin{align*} x \in L_{K'} \cap (K'^\times)^E \end{align*} with valuation vector $w$. Thus every $w \in \widetilde{\mathcal B}(M(L))$ lies in the set whose closure defines $\operatorname{Trop}(L)$. Consequently, \begin{align*} \widetilde{\mathcal B}(M(L)) \subset \operatorname{Trop}(L). \end{align*}[/step]

[guided]The remaining direction is a lifting statement: from the combinatorial condition on $w$, we have shown that the initial ideal has no monomial, and we now need an actual valued point whose coordinate valuations are $w$. This is exactly the content of the initial-ideal form of the [Fundamental Theorem of Tropical Geometry](https://doi.org/10.1090/gsm/161). In the form used here, initial forms are computed after rescaling minimal-weight terms to valuation zero and reducing their coefficients in the residue field; if $K$ is not algebraically closed, we may first pass to a valued algebraic closure, and the monomial-free condition for the initial ideal is preserved. With this convention, the theorem applies to an ideal $I \subset K[x_e : e \in E]$ and a real weight vector $w \in \mathbb{R}^E$; it says that $\operatorname{in}_w(I)$ contains no monomial if and only if, after possibly replacing $K$ by an algebraically closed valued extension $K'$ with real-valued value group containing the coordinates of $w$, there is a torus point \begin{align*} x=(x_e)_{e \in E} \in V(I)(K') \cap (K'^\times)^E \end{align*} with $v'(x_e)=w_e$ for every $e \in E$. We check the hypotheses in this situation. The ideal $I(L)$ is an ideal of the polynomial ring $K[x_e : e \in E]$, and it is homogeneous and linear because it is the ideal of linear equations cutting out the linear subspace $L \subset K^E$. The vector $w$ is real-valued by assumption, since $w \in \widetilde{\mathcal B}(M(L)) \subset \mathbb{R}^E$. The previous step proved that $\operatorname{in}_w(I(L))$ contains no monomial. Therefore the lifting direction of the fundamental theorem applies. The theorem statement defining $\operatorname{Trop}(L)$ allows valued field extensions whose value group is real-valued. This is compatible with the lifting theorem here: the extension only needs to contain the original value group and the finitely many real numbers $w_e$, so its value group can be taken inside $\mathbb{R}$. Hence we obtain a valued extension $K'/K$ and a point \begin{align*} x \in V(I(L))(K') \cap (K'^\times)^E \end{align*} with $v'(x_e)=w_e$ for all $e \in E$. Finally, $V(I(L))(K')$ is the scalar extension $L_{K'}$ because $I(L)$ is the full linear ideal of equations vanishing on $L$. Thus \begin{align*} x \in L_{K'} \cap (K'^\times)^E, \end{align*} and its valuation vector is exactly $w$. Therefore $w \in \operatorname{Trop}(L)$, and since $w \in \widetilde{\mathcal B}(M(L))$ was arbitrary, we get \begin{align*} \widetilde{\mathcal B}(M(L)) \subset \operatorname{Trop}(L). \end{align*}[/guided]

[step:Conclude equality of the tropicalization and the Bergman fan] The second step proved \begin{align*} \operatorname{Trop}(L) \subset \widetilde{\mathcal B}(M(L)), \end{align*} and the lifting argument proved \begin{align*} \widetilde{\mathcal B}(M(L)) \subset \operatorname{Trop}(L). \end{align*} Therefore \begin{align*} \operatorname{Trop}(L)=\widetilde{\mathcal B}(M(L)). \end{align*} The equality is exactly the assertion that a linear space realized over a coefficient field with zero valuation on nonzero scalars has tropicalization equal to the Bergman fan of its underlying matroid. [/step]