Tropical bases by regular projections Kerstin Hept Eindhoven, - - PowerPoint PPT Presentation

✁ ✄

Tropical bases by regular projections

Kerstin Hept

Eindhoven, 10/31/07

joint work with Thorsten Theobald www.arXiv.org/abs/0708.1727

✁

Topics

✄

First session:

• Tropical hypersurface
• Newton polytope and its subdi-

vision

• Properties of the tropical hyper-

surface

• Real valuations, Puiseux series
• Tropicalization of a polynomial
• Tropical variety and prevariety

and their properties

• Tropical bases
• Linear spaces

Second session:

• The main theorem
• Construction
• f

appropriate projections

• Preimage of a projection
• Example

✁

Tropical Semiring

✄

The main algebraic structure in tropical geometry is the tropical semiring:

• Definition. The tropical semiring is the triple (R∪{∞}, ⊕, ⊙) with the following

tropical addition and multiplication a ⊕ b := min(a, b) a ⊙ b := a + b We observe:

• The compositions are commutative, associative und distributive.
• ∞ is the additive and 0 is the multiplicative neutral element
• There is NO TROPICAL SUBTRACTION.

✁

Tropical polynomial ring

✄

Tropical polynomial ring in the unknowns x1, . . . , xn: Monomials: x⊙i1

1

⊙ x⊙i2

2

⊙ . . . ⊙ x⊙in

n

i1, . . . , in ∈ Z Polynomials: Finite linear combinations of monomials p(x1, . . . , xn) = a ⊙ xi1

1 ⊙ xi2 2 ⊙ . . . ⊙ xin n ⊕

b ⊙ xj1

1 ⊙ xj2 2 ⊙ . . . ⊙ xjn n ⊕ . . .

= min{a + i1x1 + . . . + inxn, b + j1x1 + . . . + jnxn, . . .}

✁

Tropical hypersurface

✄

• Definition. Let p(x1, . . . , xn) be a tropical polynomial. Then

T (p) := {x ∈ Rn | the minimum is attained at least twice in x} is the tropical hypersurface defined by p.

• Example. Tropical curves in the plane

p := 0 ⊙ x1 ⊕ 1 ⊙ x2 ⊕ 2

• Figure: a tropical line

q := 1⊕0⊙x1⊕0⊙x2⊕1⊙x2

1⊕0⊙x1⊙x2⊕1⊙y2

• Figure: a tropical quadratic curve

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Newton polytope

✄

• Definition. Let f be a tropical polynomial in the unknowns x1, . . . , xn with terms

xi1

1 ⊙ . . . ⊙ xin n . Then the Newton polytope of f is the convex hull

New(f) := conv{(i1, . . . , in) ∈ Rn : xi1

1 · · · xin n a term in p}

• Example. f = 2 ⊙ x2 ⊕ x2 ⊙ y3 ⊕ 3 ⊙ x ⊙ y2 ⊕ 1

✲ ✻

• ✁

✁ ✁ ✁ ✁ ✁

1 1 2 2 3 Figure: the Newton polytope of f

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Subdivision

✄

• Definition. For a tropical polyno-

mial p the extended Newton polytope is: conv{(i1, . . . , in, ci) : ci⊙xi1

1 ⊙. . .⊙xin n a monomial in p}

The projection of the lower bound

• nto the first n coordinates gives a

subdivision of the Newton polytope. The tropical hypersurface is dual to that subdivision.

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Examples

✄

• Example. A subdivided Newton polytope and its dual.

p := 3 ⊙ x2

1 ⊕ 2 ⊙ x1 ⊙ x2 ⊕ 3 ⊙ x2 2 ⊕ 0

❅ ❅ ❅ ❅ ❅

• 3

3 2

• ❅

❅ ❅

• 2

2

• Example. One can recognize the type of the tropical hypersurface only by analysing

the subdivided Newton polytope.

❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅

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Balancing condition

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Let e be an edge of a tropical curve T (f) in R2. Then one can define its multiplicity by the lattice length of the corresponding edge in the subdivided Newton polytope. Let p be any vertex of T (f), v1, . . . , vr the primitive lattice vectors in the direction

• f the edges emanating from p and m1, . . . , mr the corresponding multiplicities.

Then the following balancing condition holds: m1 · v1 + m2 · v2 + . . . + mr · vr = 0 (This holds because of the duality)

❄

• ✒

❅ ❅ ■

• 2

2 2 2 2 · (0, −1) + (1, 1) + (−1, 1) = 0

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Real valuations

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For a field K, a real valuation is a map ord : K → ¯ R = R ∪ {∞} with

• K \ {0} → R and
• 0 → ∞
• ord(ab) = ord(a) + ord(b) and
• ord(a + b) ≥ min{ord(a), ord(b)}.
• Example. K = Q with the p-adic valuation: Every q ∈ Q with

q = ps m n , p |m, p |n, s ∈ Z has the valuation

• rd(q) = vp(q) := s.

We can extend the valuation map to an algebraic closure ¯ K and then to ¯ Kn via

• rd : ¯

Kn → ¯ Rn, (a1, . . . , an) → (ord(a1), . . . , ord(an)) .

✁

Puiseux series

✄

Another interesting field is C{{t}}, the field of puiseux series: The elements are formal power series

• with coefficients in C.
• with rational exponents, bounded below with a common denominator

C{{t}} is an algebraic closed field. A valuation is given by the order map ord:

• rd : C{{t}} → R
• α∈A

cαxα → min{α | cα = 0}

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Tropicalization

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Let f =

α cαxα be a polynomial in K[x1, . . . , xn] where K is a field with a

valuation ord.

• Definition. The tropicalization of f is defined as

trop(f) :=

• α
• rd(cα) ⊙ xα

=

• α
• rd(cα) ⊙ xα1

1 ⊙ . . . ⊙ xαn n

= min

α {ord(cα) + α1x1 + · · · + αnxn}

and the tropical hypersurface of f is T (f) := T (trop(f)) = {w ∈ Rn : the minimum in trop(f) is attained at least twice in w}

✁ ✄

• Example. Let f = 2x + 4y − x2 + 3y2 a polynomial in Q[x, y] with the 2-adic
• valuation. Then

trop(f) = 1 ⊙ x ⊕ 2 ⊙ y ⊕ 0 ⊙ x2 ⊕ 0 ⊙ y2 = min{1 + x, 2 + y, 2x, 2y}

✲ ✻

• ✟✟✟✟✟

✟

• 1

1 2 2 3 3 Figure: The tropical hypersurface T (f)

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Tropical variety

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We generalize that to ideals:

• Definition. For an ideal I ✁ K[x1, . . . , xn], the tropical variety of I is defined by

T (I) =

• f∈I

T (f) There is another description of the tropical variety of an ideal: Proposition (Kapranov). If the valuation is nontrivial, then the tropical variety is the topological closure T (I) = ord V(I) where V(I) ⊂ ( ¯ K∗)n is the variety of I.

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Example

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Let I be generated by f1 := 2 + y − 4x2y + x2y2 + 2xy2 f2 := xyz − 2z + 4xyz2 − 2 + z2 Then we get the tropical variety in R3: Figure: Tropical variety T (I)

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Properties

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A tropical variety has several properties: Proposition (Bieri-Groves 1984). Let I be a prime ideal. Then T (I) is a pure m-dimensional complex where m = dim(I) is the Krull dimension of the ideal.. Proposition (Bieri-Groves 1984). T (I) is totally concave, which means that each convex hull of a local cone of a point x is an affine subspace.

❄

• ✒

❅ ❅ ❅ ■

• x
• ❅

❅ ❅ ❅ ❅

LCx(T (I)) Speyer showed in 2005 that the balancing condition, which is stronger than the concavity condition, holds for tropical varieties in general.

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Tropical prevariety

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• Definition. Let f1, . . . , fs ∈ K[x1, . . . , xn] be polynomials. Then the tropical pre-

variety defined by f1, . . . , fs is the intersection of the tropical hypersurfaces T (f1, . . . , fs) :=

s

• i=1

T (fi) This is in general not a tropical variety: Example. f1 := t + (t + 1)x2 + (2t3 − t4)y2 + txy f2 := t + (t

1 2 + t 3 2 )x + t 3 2 y

❅ ❅ ❅

• (1

2, −1 2)

(1, −1)

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Tropische Basis

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• Definition. A basis F = {f1, . . . , fs} of I is a tropical basis, if

T (I) =

s

• i=1

T(fi)

• Proposition. Let I ✁K[x1, . . . , xn] be an ideal. Then there exists a tropical basis.
• Example. Adding the following polyno-

mials to {f1, f2} we get a tropical basis: f3 := (3t − t2)x2 + (−t

1 2 + 4t 3 2

−2t

5 2 )x + (t + 2t2 − t3);

f4 := (3t3 − t4)y2 + (t

3 2 + 2t 5 2 )y

+(2t + t2)

❅ ❅ ❅

• (1

2, −1 2)

(1, −1)

• t

✁

Linear Spaces

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A tropical d-plane in Rn is a tropical variety T (I) where I =

n

• j=1

ai,jxj : i = 1, . . . , n − d. Here (ai,j) ∈ K(n−d)×n and rank(ai,j) = n − d. The Pl¨ ucker coordinates are given by Pi1...id := (−1)i1+...+id det      a1,j1 · · · a1,jn−d . . . ... . . . an−d,j1 · · · an−d,jn−d      = 0 Here 1 ≤ i1 < . . . < id ≤ n and 1 ≤ j1 < . . . < jn−d ≤ n is the complement.

• Lemma. A tropical basis for I is given by the circuits

Ci0...id =

d

• r=0

(−1)rPi0...ˆ

ir...idxir

✁

Linear Spaces

✄

Bogart, Jensen, Speyer, Sturmfels and Thomas showed:

• Proposition. For any 1 ≤ d ≤ n, there is a linear ideal I in C[x1, . . . , xn] such

that any tropical basis of linear forms in I has size at least

1 n−d+1

n

d

• .

(Here C has the trivial valuation) Summary:

• We know the tropical semiring,
• real valuation, tropicalization trop(f), tropical hypersurface T (f)
• tropical variety of an ideal T (I).
• There exist tropical bases.
• There are linear ideals where every tropical basis is big if we do not allow

higher degrees of the polynomials. But we can show that there are indeed small tropical bases if we drop this as- sumption.

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The main theorem

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Theorem (Theobald-H.). Let I ✁ K[x1, . . . , xn] be a prime ideal generated by the polynomials f1, . . . , fr. Then there exist g0, . . . , gn−dim I ∈ I with T (I) =

n−dim I

• i=0

T (gi) (1) and thus G := {f1, . . . , fr, g0, . . . , gn−dim I} is a tropical basis for I of cardinality r + codim I + 1. Because every algebraic set in n-space is the intersection of n hypersurfaces we have the corollary:

• Corollary. Let I ✁ K[x1, . . . , xn] be a prime ideal. Then there is a tropical basis

for I of cardinality at most n + codim I + 1.

✁

Topics

✄

To prove the result we need the following steps:

• Proposition of Bieri and Groves:

Let I✁K[x1, . . . , xn] be a prime ideal. Then there exist codim I+1 projections π0, . . . , πcodim I : Rn → Rdim(I)+1 such that T (I) =

codim I

• i=0

π−1

i

πi(T (I)) .

• Then we want to proof that π−1π(T (I)) is a tropical hypersurface.
• Computation of an example

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Construction of the projections

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First we want to find m := codim(I) projections π1, . . . , πcodim(I) : Rn → Rm+1 with the property, that X =

n

• i=codim(I)

π−1

i

πi(T (I)) is m-dimensional. Clearly it contains T (I). This is done inductively: Let X0 = {Rn}. Let πcodim(I) be an arbitrary projection with m-dimensional

• image. Then

X1 := {cells in the preimage π−1

codim(I)πcodim(I)(T (I))}.

So all maximal cells have dimension n − 1 and T (I) ⊂ |X1|.

✁

Construction of the projections

✄

Assume Xt is constructed and is (n − t)-dimensional with T (I) ⊂ |Xt| =

codim(I)

• i=codim(I)−t+1

π−1

i

πi(T (I)). Let B be the finite set of all (n − t)-dimensional subspaces of Rn parallel to at least one of the affine subspaces supporting cells of Xt. Then choose πcodim(I)−t : Rn → Rm+1 as a projection with Rn = ker πcodim(I)−t + V for all V ∈ B. Let Y be the set of all (n−1)-dimensional subspaces of π−1

codim(I)−tπcodim(I)−t(T (I))

and let Xt+1 = {Y ∩ X | Y ∈ Y, X ∈ Xt} Now Xt+1 has dimension n − t − 1 and T (I) ⊂ |Xt+1|. Choose X := Xcodim(I).

✁

Construction of the projections

✄

• Definition. Let C be a polyhedral complex in Rn. A projection π : Rn → Rm+1

is called geometrically regular if the following two conditions hold.

• 1. For any k-face σ of C we have dim(π(σ)) = k, 0 ≤ k ≤ dim C .
• 2. If π(σ) ⊆ π(τ) then σ ⊆ τ for all σ, τ ∈ C .

Secondly choose π0 to be a geometrically regular projection with respect to X. Then the m-dimensional cells of X are mapped to m-dimensional cells of Rm+1 and two different m-dimensional cells have different images. So a m-dimensional cell C is in T (I) iff π−1

0 π0(C) is in π−1 0 π0(T (I)).

T (I) is a pure m-dimensional polyhedral complex so T (I) =

codim I

• i=0

π−1

i

πi(T (I)) .

✁

Preimage is a hypersurface

✄

Let m = dim(I) and the projection be described by π : Rn → Rm+1 , x → Ax with a regular rational matrix A whose rows are denoted by a(1), . . . , a(m+1).

• Example. Here: n = 3, m = 1, l = 1

❍ ❍ ✟ ✟

Projection on the x − y-plain

Let u(1), . . . , u(l) ∈ Qn with l := n − (m + 1) be a basis of the orthogonal comple- ment of span{a(1), . . . , a(m+1)}.

✁

Preimage is a hypersurface

✄

Set R = K[x1, . . . , xn, λ1, . . . , λl], and define the ideal J ✁ R by J =

• g ∈ R : g = f(x1

l

• j=1

λju(j)

1 , . . . , xn

l

• j=1

λju(j)

n ) for some f ∈ I

• .
• Theorem. Let I ✁ K[x1, . . . , xn] be an m-dimensional prime ideal and π : Rn →

Rm+1 be a rational projection. Then π−1(π(T (I))) is a tropical variety with π−1(π(T (I))) = T (J ∩ K[x1, . . . , xn]) . (2) T (J ∩K[x1, . . . , xn]) is a tropical hypersurface if the projection is m-dimensional. But there can occur some degenerations:

P P P ❅ ❅ ❅❍❍ ❍ ❍ ❍ ✟ ✟ ❅ ❅ ❅ ❍❍ ❍

✁

Preimage is a hypersurface

✄

• Lemma. For any w ∈ T (J ∩ K[x1, . . . , xn]) and u ∈ span{u(1), . . . , u(l)} we have

w + u ∈ T (J ∩ K[x1, . . . , xn]).

• Proof. Let w.l.o.g. u = l

i=1 µju(j) with µ1, . . . , µl ∈ Q.

Let w ∈ T (J ∩ K[x1, . . . , xn]). Then we can assume w.l.o.g that there exists z ∈ V(J ∩ K[x1, . . . , xn]) with ord z = w. Define y = (y′, y′′) ∈ ( ¯ K∗)n+l by y = (y′, y′′) =

• z1t

Pl

j=1 µju(j) 1 , . . . , znt

Pl

j=1 µju(j) n , t−µ1, . . . , t−µl

• .

For any f ∈ I, the point y is a zero of the polynomial f(x1

l

• j=1

λju(j)

1 , . . . , xn

l

• j=1

λju(j)

n ) ∈ R ,

and thus y ∈ V(J). Hence, y′ ∈ V(J ∩ K[x1, . . . , xn]). Moreover,

• rd y′ = (w1 +

l

• j=1

µju(j)

1 , . . . , wn + l

• j=1

µju(j)

n ) = w + l

• j=1

µju(j) = w + u .

✁

Preimage is a hypersurface

✄

• Lemma. Let I ✁ K[x1, . . . , xn] be an ideal. Then J ∩ K[x1, . . . , xn] ⊆ I.
• Proof. Let p =

i higi be a polynomial in J ∩ K[x1, . . . , xn] with

gi = fi(x1

l

• j=1

λju(j)

1 , . . . , xn

l

• j=1

λju(j)

n ) ∈ R and fi ∈ I .

Since p is independent of λ1, . . . , λl we have p = p|λ1=1,...,λl=1 =

• i

hi|λ1=1,...,λl=1 fi ∈ I.

• Definition. We call a projection algebraically regular for I if for each i ∈ {1, . . . , l}

the elimination ideal J ∩ K[x1, . . . , xn, λ1, . . . , λi] has a finite basis Fi such that in every polynomial f ∈ Fi the coefficients of the powers of λi (when considering f as a polynomial in λi) are monomials in x1, . . . , xn, λ1, . . . , λi−1.

✁

Preimage is a hypersurface

✄

• Theorem. Let I ✁ K[x1, . . . , xn] be a prime ideal and π : Rn → Rm+1 be an

algebraically regular projection. Then π−1π(T (I)) is a tropical variety with π−1π(T (I)) = T (J ∩ K[x1, . . . , xn]) . (3)

• Proof. Idea:

Let w ∈ π−1π(T (I)). w.l.o.g. it exists z′ ∈ V(I) and u ∈ span{u(1), . . . , u(l)} with ord z′ = w + u. For any f ∈ I, the point z := (z′, 1) is in the variety V(J). Hence, z′ ∈ V(J ∩ K[x1, . . . , xn]) By Lemma 1, w ∈ T (J ∩ K[x1, . . . , xn]) as well.

✁

Preimage is a hypersurface

✄

Let now w ∈ T (J ∩ K[x1, . . . , xn]). Again we can assume that there is a z ∈ V(J ∩ K[x1, . . . , xn]) ⊆ (K∗)n with w = ord(z). The projection is algebraically regular so by the Extension Theorem, we can extend the root z inductively to a root ˜ z ∈ V(J) with the same first n

• entries. The definition of J says that

z′ := (z1˜ zu(1)

1

n+1 · · · ˜

zu(l)

1

n+l, . . . , zn˜

zu(1)

n

n+1 · · · ˜

zu(l)

n

n+l)

is a root of I. Then

• rd(z′) = ord(z) +

l

• i=1
• rd(˜

zn+i)u(i) which means that ord(z) = w ∈ π−1π(T (I)).

• Remark. We can show, that we can always lift a point in the tropical variety

T (J ∩ K[x1, . . . , xn]) to a point in the tropical variety T (J), so the theorem holds in general.

✁

Example

✄

Example. Let I ✁ Q[x, y, z] be generated by f1 := 2 + y − 4x2y + x2y2 + 2xy2 f2 := xyz − 2z + 4xyz2 − 2 + z2 and let ord be the 2-adic valuation. For the first projection π2 we take the one with kernel (0, 0, 1), so it is the projec- tion on the plane z = 0. Then J ∩ K[x, y, z] is generated by f1 and the tropical variety is

V1 := {(x, y, z) ∈ R3 | x = 1/2 − y, 1 ≤ y ≤ 2, z ∈ R}, V2 := {(x, y, z) ∈ R3 | y = 1, −1/2 ≤ x, z ∈ R}, V3 := {(x, y, z) ∈ R3 | x = 1, y ≤ −2, z ∈ R}, V4 := {(x, y, z) ∈ R3 | y = −1 − 2x, x ≤ −3/2, z ∈ R}, V5 := {(x, y, z) ∈ R3 | x = −1 − y, y ≤ −2, z ∈ R}, V6 := {(x, y, z) ∈ R3 | y = −2x, −1/2 ≤ x ≤ 1, z ∈ R}, V7 := {(x, y, z) ∈ R3 | y = 2, x ≤ −3/2, z ∈ R}.

✁

Example

✄

The kernel of the second projection π1 should not lie in the subspaces parallel to the supporting affine subspaces of the above sets. We can choose for example (1, 1, 0). J ∩ K[x, y, z] = 192xyz + 1008xyz2 + 16xy + 2176xyz3 + 1996xyz4 + 448z5xy −260z6xy + 1153xyz8 + 712xyz7 − 128x2z − 32y2z − 1728x2z3 −896x2z2 − 512x2z4 − 594y2z3 − 240y2z2 − 666y2z4 − 368x2z7 +288x2z6 + 1120x2z5 + 64x2z8 + 52y2z7 − 16y2z6 − 335y2z5 + 16y2z8 The ideal is generated by our third polynomial f3. This gives us a 1-dimensional set X, which consists of the supporting affine subspaces of the prevariety T (f1) ∩ T (f3).

✁

Example

✄

To choose a geometrically regular projection with respect to X we can for example take (2, 4, 1) as a kernel for π0. Computing the polynomial in the elimination ideal (it has 63 terms) and the intersection of the tropical variety of all three polynomials we get:

✁

References

✄

References

[1] R. Bieri and J.R.J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347 (1984), 168–195. [2] M. Einsiedler, M.M. Kapranov, and D. Lind, Non-archimedean amoebas and tropical varieties, J. Reine Angew. Math. 601 (2006), 139–157. [3] D. Eisenbud and E.G. Evans, Every algebraic set in n-space is the intersection

• f n hypersurfaces, Inventiones math. 19 (1973), 107–112.

[4] K. Hept and T. Theobald, Tropical bases by regular projections, Preprint, arXiv:0708.1727, 2007. [5] J. Richter-Gebert, B. Sturmfels, and T. Theobald, First steps in tropical ge-

• metry, Idempotent Mathematics and Mathematical Physics, Contemp. Math.

377 (2005), 289–317. [6] D. Speyer, Tropical geometry, Ph.D. thesis, Harvard University, 2005.