Tropical Discriminants Eva Maria Feichtner feichtne@math.ethz.ch - - PowerPoint PPT Presentation

Tropical Discriminants

Eva Maria Feichtner

feichtne@math.ethz.ch

Department of Mathematics, ETH Zurich

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.1/25

Outline

1. A-Discriminants 2. Tropical Geometry 3. Tropical A-Discriminants 4. The Newton Polytope of ∆A 5. Regular Subdivisions and ∆-Equivalence of Triangulations joint work/project with Alicia Dickenstein and Bernd Sturmfels

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.2/25

1.

A-Discriminants

[Gelfand, Kapranov, Zelevinsky 1992]

A =

• a1 · · · an
• ∈ Zd×n, rk A = d,

(1, . . . , 1) ∈ row spanA QA = conv {a1, . . . , an} polytope in Rd, dim QA = d − 1 XA = V ( xu − xv | u, v ∈ Nn with Au = Av ) projective toric variety X∗

A = cl {ξ ∈ (CPn−1)∗ | Hξ tangent to XA at a regular point}

dual variety If codim X∗

A = 1,

X∗

A = V(∆A),

where ∆A is a unique irreducible polynomial, the A-discriminant.

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.3/25

A-Discriminants: Classical Examples

• 1. Discriminant of a quadratic polynomial in 1 variable

f(x) = a2x2 + a1x + a0, a2 = 0 f has a double root ⇐ ⇒ ∆f = a2

1 − 4a2a0 = 0

∆f = ∆A ∈ Z[a0, a1, a2] for A =

• 1

1 1 1 2

• 2. Discriminant of a degree n polynomial in 1 variable

f(x) = n

i=0 aixi,

an = 0 f has a double root ⇐ ⇒ ∆f = 0 ∆f = ∆A ∈ Z[a0, . . . , an] for A =

• 1

1 . . . 1 1 . . . n

• E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.4/25

A-Discriminants: Classical Examples

• 2. Resultant of two polynomials in 1 variable

f(x) =

n

• i=0

aixi, an = 0, g(x) =

m

• i=0

bixi, bm = 0, f and g have a common root ⇐ ⇒ Res(f, g) = 0 Res(f, g) = ∆A ∈ Z[a0, . . . , an, b0, . . . , bm] for A =    1 1 . . . 1 . . . . . . 1 1 . . . 1 1 . . . n 1 . . . m    Res(f, g) = determinant of the Sylvester matrix

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.5/25

A-Discriminants: Classical Examples

• 3. Discriminant of a deg 2 homogeneous polynomial in 3 variables

A =    2 1 1 1 2 1 1 1 2    ∆A = det    2a1 a2 a3 a2 2a4 a5 a3 a5 2a6   

• 4. Discriminant of a deg 3 homogeneous polynomial in 3 variables

A =    1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 1 2 3 1 2 1    deg ∆A = 12, 2040 terms

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.6/25

A-Discriminants

Call A defective if codim X∗

A > 1 .

The dual variety X∗

A is also of interest in the defective case.

Goal: Derive information on ∆A, resp. X∗

A, for instance

degree and extreme monomials of ∆A dimension, degree and Chow form of X∗

A

directly from A, without any reference to defining equations. Ansatz: Study the tropicalization of X∗

A !

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.7/25

• 2. Tropical Geometry

(R ∪ {∞}, ⊕, ⊗), x ⊕ y := min{x, y} , x ⊗ y := x+y tropical semi-ring complex projective varieties

τ

− → polyhedral fans Y ⊆ CPn−1 irreducible variety, dim Y = r IY ⊆ C[x1, . . . , xn] defining prime ideal τ(Y ) = { w ∈ Rn | inw(IY ) does not contain a monomial } tropicalization of Y τ(Y ) is a pure r-dimensional polyhedral fan in Rn, respectively TPn−1 = Rn/ R(1, 1, . . . , 1).

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.8/25

Examples of Tropicalized Varieties

• 1. Y hypersurface in CPn−1

f ∈ C[x1, . . . , xn] irreducible polynomial defining Y New(f) Newton polytope, NNew(f) its normal fan τ(Y ) = codim 1-skeleton of NNew(f)

Proof:

τ(Y ) = { w ∈ Rn | inw(f) is not a monomial } = { w ∈ Rn | dim

• New(inw(f))
• > 0}

= { w ∈ Rn | dim

• w-maximal face of New(f)
• > 0}

=

• σ∈NNew(f)

codim σ>0

σ

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.9/25

Examples of Tropicalized Varieties

• 2. Y = XA toric variety, A ∈ Zd×n

τ(Y ) = row span A

Proof:

IXA = xu − xv | u, v ∈ Nn with Au = Av τ(Y ) = { w ∈ Rn | inw(f) is not a monomial for any f ∈ IXA} = { w ∈ Rn | wu = wv whenever Au = Av} =

row span A

• 3. Y = V linear, resp. projective subspace

τ(Y ) = B(M(V )) Bergman fan of the matroid associated with V

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.10/25

Digression: Bergman Fans of Matroids

M connected matroid on {1, . . . , n}, rk M = r Mw = {σ ∈ M | σ basis with maximal w-cost } for w ∈ Rn B(M) = {w ∈ Rn | Mw is loop-free } Bergman fan B(M) = B(M) ∩

• w ∈ Rn
• wi = 0,
• w2

i = 1

• Bergman complex

B(M) is a (rk M − 1)-dimensional subfan of NP(M), where P(M) is the matroid polytope of M.

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.11/25

Examples of Bergman Fans

M = M(K4) r = 3, n = 6

2 5 1 4 3 6 K4 1 3 236 5 6 146 125 345 2 4 B(M(K4))

M = M(K4\e) r = 3, n = 5

125 3 4 2 1 345 5 1 4 3 2 B(M(K4\e)) K4\e

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.12/25

Digression: Nested Set Fans of Matroids

M connected matroid on {1, . . . , n}, rk M = r LM lattice of flats G ⊆ (LM)>ˆ

0 building set if for any X ∈ LM and

max G≤X = {G1, . . . , Gk}, there exists an isomorphism φX :

k

• i=1

[ˆ 0, Gi] − → [ˆ 0, X] . Gmin: irreducibles, dense edges, connected flats Gmax: LM \ {ˆ 0} S ⊆ G nested set if for any pairwise incomparable X1, . . . , Xt ∈ S, t ≥ 2, Xi ∈ G. N(G) abstract simplicial complex of nested sets N (G) realization as a simplicial fan in Rn

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.13/25

Examples of Nested Set Fans

M = M(K4) r = 3, n = 6

1 3 236 5 6 146 125 345 2 4 N(Gmin)

1 2 3 4 5 6

1 3 236 5 6 146 125 345 2 4 N(Gmax)

LM

M = M(K4 \ e) r = 3, n = 5

125 345 N(Gmin) 125 3 4 2 1 345 3 4 2 1 N(Gmax)

1 2 4 3 5 L′

M

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.14/25

Bergman Fans versus Nested Set Fans

Proposition:

[F. & Sturmfels ’04; F. & M¨ uller ’03]

B(M) is subdivided by N (G) for any building set G in LM. N (G) is subdivided by N (G′) for any building sets G ⊆ G′ in LM. Back to tropical geometry: V a linear subspace in Cn, M the associated matroid, G any building set in LM τ(V ) =

supp B(M) = supp N (G)

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.15/25

3. Tropical A-Discriminants

A =

• a1 · · · an
• ∈ Zd×n, rk A = d,

(1, . . . , 1) ∈ row spanA Horn uniformization of A-discriminants:

[Kapranov ’91]

The dual variety X∗

A is the closure of the image of the morphism

ϕA : P(ker A) × (C∗)d/C∗ − → (CPn−1)∗ (u, t) − → (u1ta1 : u2ta2 : · · · : untan) . Tropical Horn uniformization: τ(ϕA) : B(ker A) × Rd − → TPn−1 (w, v) − → w + vA im τ(ϕA) = B(ker A) + row span A Horn fan

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.16/25

Tropical A-Discriminants

Theorem: [DFS ’05] τ(X∗

A) = B(ker A) + row span A

Example: A =    2 1 1 1 2 1 1 1 2   

245 123 2 356 5 4 1 6 3 2 3 5 4 1 6

B(ker A) τ(X∗

A)

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.17/25

Tropicalizing Monomials in Linear Forms

f : Cm

U

− → Cr

V

− → Cs U ∈ Cr×m linear map, V ∈ Zs×r monomial map fi(x1, . . . , xm) =

r

• k=1

(uk1x1 + · · · + ukmxm)vik YUV := closure of im f Examples: r = s, V = Ir: YUV = im U linear space m = r, U = Im: YUV = XV T toric variety Theorem: [DFS ’05] τ(YUV ) = V ◦ τ(im U) = V ◦ B(M(im U))

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.18/25

Tropicalizing Monomials in Linear Forms

Retrieving the tropical discriminant: Set m = r, r = n + d, s = d, B a Gale dual of A, U =

• B

Id

• and

V =

• In

AT . Then, Y P

UV = X∗ A , and

τ(X∗

A) = B(ker A) + row span A .

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.19/25

4. The Newton Polytope of ∆A

A ∈ Zd×n non-defective, L lattice of flats of M(ker A), N any nested set fan of L, σ ∈ N ← → σ1, . . . , σn−d−1 ∈ {0, 1}n w ∈ Rn generic, i ∈ {1, . . . , n}, Ni,w = {σ ∈ N | row spanA ∩ R>0{σ1, . . . , σn−d−1, −w, −ei} = ∅ }. Theorem: [DFS ’05] deg xi

• inw(∆A)
• =
• σ∈Ni,w
• det
• AT , σ1, . . . , σn−d−1, ei

, in fact, deg xi

• inw(∆A)
• is the number of intersection points of

w + R>0ei with τ(X∗

A), counted with multiplicity.

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.20/25

The Newton Polytope of ∆A

Example: A =    2 1 1 1 2 1 1 1 2   

2 3 5 4 1 6

τ(X∗

A)

New(∆A)

a2

3a4

a2

2a6

a1a4a6 a1a2

5

a2a3a5

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.21/25

• 5. Regular Subdivisions

A =

• a1 · · · an
• ∈ Zd×n, point configuration a1, . . . , an in Zd

For w ∈ Rn, Πw :=

• regular subdivision of A given by lower facets of

conv {(a1, w1), . . . , (an, wn)} ⊆ Rd × R . For σ a maximal cell in Πw, M(σ) the associated rk d matroid, define the affine linear function ψσ : Rd → R by ψσ(a) =

• 1 ,

if A is a co-loop in M(σ) ,

0 ,

• therwise .

and set σ∗ := σ ∪ {a ∈ A | ψσ(a) < 1} ⊆ A. Call a ∈ σ a strong co-loop if it is a co-loop in M(σ∗).

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.22/25

Regular Subdivisions

Theorem: [DFS ’05] The tropical A-discriminant τ(X∗

A) equals

{ w ∈ Rn | Πw has a maximal cell σ with no strong co-loops }. Corollary: τ(X∗

A) is a subfan of the secondary fan Σ(A).

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.23/25

∆-Equivalence of Triangulations

For a non-defective point configuration A, two regular triangulations Πw and Πw′ are called ∆-equivalent if inw(∆A) = inw′(∆A) . Corollary: Two neighboring triangulations Πw and Πw′ are ∆-equivalent if and only if every cell in their coarsening has a strong co-loop.

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.24/25

Example

A =    2 1 1 1 2 1 1 1 2    τ(X∗

A) ⊆ Σ(A)

E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.25/25