Tropical aspects of algebraic matroids. Jan Draisma Universität Bern and TU Eindhoven (w/ Rudi Pendavingh and Guus Bollen TU/e)

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1 1 Tropical aspects of algebraic matroids Jan Draisma Universität Bern and TU Eindhoven (w/ Rudi Pendavingh and Guus Bollen TU/e)

2 Two advertisements 2 SIAGA: SIAM AG 19: 9 13 July 2019, Bern

3 Algebraic matroids 3 K algebraically closed, X K E irreducible algebraic matroid M(X) on E: I E independent : X K I dominant x + z Example (Fano and non-fano): K = F 2, X = im[k 3 K 7, (x, y, z) (x, y, z, y + z, x + z, x + y, x + y + z)] yz xz xy xyz {algebraic matroids} is closed under deletion and contraction BIG OPEN QUESTIONS: is algebraic realisability decidable? is the class closed under duality? how many algebraic matroids are there? xz x z x + y xy yz y + z y

4 Application: matrix completion 4 General observation: I independent Trop(X) R I surjective. X m,n,r = {matrices of rank r} K m n I [m] [n] independent a general partial I-matrix /K can be completed to a rank- r-matrix. r = 1 : t 1 l 1 t 1 l 2 t 2 l 2 t 2 l 3 t 3 l 1 t 3 l 3 t 1 l 1 t 1 l 2 t 2 l 2 t 2 l 3 t 3 l 3 t 3 l 4 Open: can independence in M(X m,n,r ) be tested in poly time? Theorem (Daniel Bernstein): For r = 2, I is independent iff I has an acyclic orientation without alternating cycles. Proof uses Trop(Gr 2,m+n ) Trop({skew-symmetric matrices}).

5 Characteristic zero vs nonzero 5 Lemma (Ingleton) If chark = 0, then {matroids algebraic/k}={matroids linear/k}. Proof: For general v X, M(T v X) = M(X). answers in char 0: yes, yes, 0 percent (Nelson, 2017). Does not work in char p > 0: Example X = {(t, t p ) t K}, M(X) has bases {1}, {2} but T v X = (1, 0) so M(T v X) only has basis {1}. Way out (Lindström): replace X by {(x p 1, x 2) x X} = Y = {(t, t) t K} and T v Y = (1, 1) with M(T v Y) = M(Y) = M(X).

6 An algebraic but nonlinear matroid 6 only linear in char 2 not linear in char 2 z xz yz x + z y + z x xy x + y y M(X) is a nonlinear but algebraic matroid cannot find a Y K 10 with v Y such that M(T v Y) = M(X) Way out: Frobenius flocks!

7 Frobenius flocks 7 F : a a p the Frobenius automorphism Z E acts on K E via αv := (F α i v i ) i E by Zariski-homeomorphisms For X K E and α Z E have M(X) = M(αX). Theorem (Bollen Draisma Pendavingh) For general v X, the map V : Z E Gr(d, K E ), V(α) = T αv αx has the following properties: (FF1) V α e i = diag(1,..., 1, 0, 1,..., 1)V α+ei (FF2) V α+1 = 1V α and moreover Bases(M(X)) = α Bases(M(V(α))). Definition A map V satisfying (FF1), (FF2) is called a Frobenius flock.

8 Example 1 8 (FF1) V α e i = diag(1,..., 1, 0, 1,..., 1)V α+ei (FF2) V α+1 = 1V α (X =)V 0 = (1, a), a 0 V 0 e 1 = {(0, 0)} = diag(0, 1)V e 1 V e1 = (1, 0) (1, a p2 ) FF2 yields: (0, 1) (1, a) (1, 0) (1, 0) (1, 0)

9 Example 2 9 X = {(x, y, x + y, x + y (pg) ) (x, y) K 2 } K 4, g > 1, M(X) = U 2,4 [ ] T 0 X = row space of, so 1, 4 parallel in M(T X). ( e 2 e 3 )X = {(x, y, x p + y, x + y (pg 1) ) (x, y) K 2 } [ ] T 0 ( e 2 e 3 )X = row space of ; also 2, 3 parallel ( ge 2 ge 3 )X = {(x, y, x (pg) + y, x + y) (x, y) K 2 } T 0 ( ge 2 ge 3 )X = row space of [ ] ; 1, 4 indep.

10 Example 2, continued 10 Cells where M(T αx (αx)) is constant: These cells are alcoved polytopes: max-plus and min-plus closed.

11 Matroid flocks from valuations and vice versa 11 Definition (Dress-Wenzel) A matroid valuation is a map ν : {d-sets in E} R { } such that ν(b) for some B and B, B, i B \ B j B \ B : ν(b) + ν(b ) ν(b i + j) + ν(b + i j). (ν then lies in the Dressian and defines a tropical linear space) Observations ν two matroids: M ν := {B ν(b) < } and {B ν(b) minimal}; and ν (B) := ν(b) α e B is a valuation for each α R E. Theorem (Bollen-Draisma-Pendavingh) Given a Z { }-valued ν, set M ν α := {B ν(b) α e B minimal} for each α Z E. This satisfies matroid analogues of FF1,FF2. Conversely, each such matroid flock arises in this manner.

12 Overview 12 (X, v) (α T αv αx) {algebraic varieties X K E } {Frobenius flocks V : α V α } X M(X) V (α M(V α )) {Matroids on E} M α{bases of M α } {Matroid flocks M : α M α } ν M ν Murota thanks to Yu! {Z { }-valued matroid valuations} So to a d-dimensional algebraic variety X K E in char p we associate the Lindstrom valuation ν X : {d-subsets of E} Z { }. Cartwright found a direct construction of ν X.

13 Monomial parameterisations 13 ϕ : (K ) d (K ) n monomial map, ϕ(t) = (t Ae 1, t Ae 2,..., t Ae n ), where A Z d n. Set X := imϕ. Theorem: ν X sends B [n], B = d to the p-adic valuation of det A[B]. Generalises? G a 1-dimensional algebraic group defined over F p. E := End(G) has F E. A E d n a d-dimensional subgroup X G n Theorem (I think): ν X (B) = number of factors F in the Smith normal form of A.

14 Rigidity 14 Definition (Dress-Wenzel) A matroid M is rigid if every valuation ν with M ν = M is of the form M R, B α e B for some α R E. Theorem A rigid matroid is algebraically representable over an algebraically closed field K of positive characteristic if and only if it is linearly representable over K. Proof If X is an algebraic representation, then the Lindström valuation ν X : M(X) Z sends B α e B for some α Z E. Then Mα ν = M ν. Now M(X) = M(T αx αx) for x X general. Applies to projective planes over finite fields!

15 Properties of Frobenius flocks 15 Frobenius flocks... have deletion/contraction are almost preserved under duality (replace F by F 1 ) allow for circuit hyperplane relaxations so Vamos is Frobenius flock realisable (and many more!): arxiv: (Adv. Math. 323, 2018) Thank you!