Tropical Linear Algebra Jorge Alberto Olarte February 18, 2019 Joint work with Alex Fink, Benjamin Schr¨oter and Marta Panizzut Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 1 / 14 2 As the solution set of n − d linear equations. This can be represented by a matrix A? 2 K(n−d)×n where rows give the coefficients of the linear equations. 3 By its Plucker¨ coordinates. Classical linear spaces Let K be any field. A d-dimensional linear subspace L ⊆ Kn can be given in several forms: d 1 As a span of vectors d vectors v1;:::; vd 2 K . This can be represented by a d×n matrix A 2 K where the rows are given by the vectors v1;:::; vd . Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 2 / 14 3 By its Plucker¨ coordinates. Classical linear spaces Let K be any field. A d-dimensional linear subspace L ⊆ Kn can be given in several forms: d 1 As a span of vectors d vectors v1;:::; vd 2 K . This can be represented by a d×n matrix A 2 K where the rows are given by the vectors v1;:::; vd . 2 As the solution set of n − d linear equations. This can be represented by a matrix A? 2 K(n−d)×n where rows give the coefficients of the linear equations. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 2 / 14 Classical linear spaces Let K be any field. A d-dimensional linear subspace L ⊆ Kn can be given in several forms: d 1 As a span of vectors d vectors v1;:::; vd 2 K . This can be represented by a d×n matrix A 2 K where the rows are given by the vectors v1;:::; vd . 2 As the solution set of n − d linear equations. This can be represented by a matrix A? 2 K(n−d)×n where rows give the coefficients of the linear equations. 3 By its Plucker¨ coordinates. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 2 / 14 (n)−1 A vector W 2 PK d is the vector of Plucker¨ coordinates of a linear space if and only if it satisfies the Plucker¨ relations: [n] [n] X 8S 2 ; 8T 2 ; W · W = 0 d − 1 d + 1 T ni S[i i2T nS The linear space L can be recovered from the Plucker¨ coordinates: ( ) [n] X L = x 2 n j 8T 2 ; W · x = 0 K d + 1 T ni i i2T Classical Plucker¨ coordinates [n] d×d For any B 2 d (a subset of size d of f1;:::; ng) let AB 2 K be the submatrix of A consisting of the columns indexed by B. Definition (n)−1 The Plucker¨ coordinates of L consist in a vector in PK d whose entries are the maximal minors [det(AB )] [n] of any matrix A whose rows form a basis of L. B2( d ) This does not depend on the choice of basis. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 3 / 14 The linear space L can be recovered from the Plucker¨ coordinates: ( ) [n] X L = x 2 n j 8T 2 ; W · x = 0 K d + 1 T ni i i2T Classical Plucker¨ coordinates [n] d×d For any B 2 d (a subset of size d of f1;:::; ng) let AB 2 K be the submatrix of A consisting of the columns indexed by B. Definition (n)−1 The Plucker¨ coordinates of L consist in a vector in PK d whose entries are the maximal minors [det(AB )] [n] of any matrix A whose rows form a basis of L. B2( d ) This does not depend on the choice of basis. (n)−1 A vector W 2 PK d is the vector of Plucker¨ coordinates of a linear space if and only if it satisfies the Plucker¨ relations: [n] [n] X 8S 2 ; 8T 2 ; W · W = 0 d − 1 d + 1 T ni S[i i2T nS Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 3 / 14 Classical Plucker¨ coordinates [n] d×d For any B 2 d (a subset of size d of f1;:::; ng) let AB 2 K be the submatrix of A consisting of the columns indexed by B. Definition (n)−1 The Plucker¨ coordinates of L consist in a vector in PK d whose entries are the maximal minors [det(AB )] [n] of any matrix A whose rows form a basis of L. B2( d ) This does not depend on the choice of basis. (n)−1 A vector W 2 PK d is the vector of Plucker¨ coordinates of a linear space if and only if it satisfies the Plucker¨ relations: [n] [n] X 8S 2 ; 8T 2 ; W · W = 0 d − 1 d + 1 T ni S[i i2T nS The linear space L can be recovered from the Plucker¨ coordinates: ( ) [n] X L = x 2 n j 8T 2 ; W · x = 0 K d + 1 T ni i i2T Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 3 / 14 The tropical projective space is TPn−1 = (Tn n f(1;:::; 1)g) = ∼ where (x1;:::; xn) ∼ (y1;:::; yn) if and only if there exists λ 2 T such that (x1;:::; xn) = (y1 λ, : : : ; yn λ) A tropical polynomial is of the form M α f (x1;:::; xn) = cα x α2∆ = min (cα + α1 · x1 + ··· + αn · xn) α2∆ n where the coefficients cα 2 T are tropical numbers and ∆ is a finite subset of N . The tropical analogous of the 'zeros' of f consists of al x 2 Tn where the minimum in f is either 1 or achieved twice. The tropical semiring The tropical semiring is T = (R [ f1g; ⊕; ; 1; 0) where a ⊕ b = min(a; b) a b = a + b Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 4 / 14 A tropical polynomial is of the form M α f (x1;:::; xn) = cα x α2∆ = min (cα + α1 · x1 + ··· + αn · xn) α2∆ n where the coefficients cα 2 T are tropical numbers and ∆ is a finite subset of N . The tropical analogous of the 'zeros' of f consists of al x 2 Tn where the minimum in f is either 1 or achieved twice. The tropical semiring The tropical semiring is T = (R [ f1g; ⊕; ; 1; 0) where a ⊕ b = min(a; b) a b = a + b The tropical projective space is TPn−1 = (Tn n f(1;:::; 1)g) = ∼ where (x1;:::; xn) ∼ (y1;:::; yn) if and only if there exists λ 2 T such that (x1;:::; xn) = (y1 λ, : : : ; yn λ) Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 4 / 14 The tropical semiring The tropical semiring is T = (R [ f1g; ⊕; ; 1; 0) where a ⊕ b = min(a; b) a b = a + b The tropical projective space is TPn−1 = (Tn n f(1;:::; 1)g) = ∼ where (x1;:::; xn) ∼ (y1;:::; yn) if and only if there exists λ 2 T such that (x1;:::; xn) = (y1 λ, : : : ; yn λ) A tropical polynomial is of the form M α f (x1;:::; xn) = cα x α2∆ = min (cα + α1 · x1 + ··· + αn · xn) α2∆ n where the coefficients cα 2 T are tropical numbers and ∆ is a finite subset of N . The tropical analogous of the 'zeros' of f consists of al x 2 Tn where the minimum in f is either 1 or achieved twice. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 4 / 14 Tropical linear spaces Definition (n)−1 A valuated matroid W is a vector in PT d that satisfies the 3-term tropical [n] Plucker¨ relations: for every S 2 d−2 and i; j; k; l 2 [n] n S, the minimum of (WSij WSkl ) ⊕ (WSij WSkl ) ⊕ (WSij WSkl ) is achieved twice. The tropical linear space associated to W is ( ) [n] M L := x 2 n j 8T 2 ; the minimum W · x is achieved twice T d + 1 T ni i i2T Tropical linear spaces are polyhedral complexes. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 5 / 14 Matroid polytopes Definition n The hypersimplex ∆d;n is the intersection of unit hypercube [0; 1] with the hyperplane fx1 + ··· + xn = dg.A matroid polytope M is a subpolytope of ∆d;n such that all of its edges are also edges of ∆d;n The hypersimplex ∆2;4 This square pyramid is This symplex is NOT a is an octahedron. matroid polytope. matroid polytope. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 6 / 14 Regular subdivisions Definition Given a polytope P with vertex set V and a height function h : V ! R the lower faces of conv(f(v; h(v)) 2 Rn+1 j v 2 V g) project onto P to form a polyhedral subdivision Sub(h). Such subdivisions are called regular. This gives RV a fan structure Sec(P) called the secondary fan of P, which consists of cones σ(S) = fh 2 RV j Sub(h) = Sg for each regular subdivision S. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 7 / 14 The tropical linear space L associated to W is dual to the subcomplex of Sub(W) which consists of all polytopes of Sub(W) that are not contained in any of the coordinate hyperplanes fxi = 0g Matroid subdivisions Theorem (Speyer) (n)−1 A vector W 2 PT d is a valuated matroid if and only if the polytope conv(feB j WB < 1g) is a matroid polytope and the regular subdivision Sub(W) consists of only matroid polytopes. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 8 / 14 Matroid subdivisions Theorem (Speyer) (n)−1 A vector W 2 PT d is a valuated matroid if and only if the polytope conv(feB j WB < 1g) is a matroid polytope and the regular subdivision Sub(W) consists of only matroid polytopes.