Implicitization of surfaces via Geometric Tropicalization Mar´ ıa Ang´ elica Cueto Columbia University Minisymposia on Tropical Geometry SIAM AG 2 Meeting 2011 NCSU, Raleigh October 7th 2011 Three references: Sturmfels, Tevelev, Yu: The Newton polytope of the implicit equation (2007) Sturmfels, Tevelev: Elimination theory for tropical varieties (2008) MAC: arXiv:1105.0509 (2011) (and many, many more!) M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 1 / 10
Implicitization problem: Classical vs. tropical approach Input: Laurent polynomials f 1 , f 2 , . . . , f n ∈ C [ t ± 1 1 , . . . , t ± 1 d ]. Algebraic Output : The prime ideal I defining the Zariski closure Y of the image of the map: f = ( f 1 , . . . , f n ): T d ��� T n The ideal I consists of all polynomial relations among f 1 , f 2 , . . . , f n . Existing methods: Gr¨ obner bases and resultants. GB: always applicable, but often too slow. Resultants: useful when n = d + 1 and I is principal , with limited use. Geometric Output : Invariants of Y , such as dimension, degree, etc. Punchline: We can effectively compute them using tropical geometry. TODAY: Study the case when d = 2 and Y is a surface . M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 2 / 10
Example: parametric surface in T 3 Input : Three Laurent polynomials in two unknowns: x = f 1 ( s , t ) = 3 + 5 s + 7 t , y = f 2 ( s , t ) = 17 + 13 t + 11 s 2 , z = f 3 ( s , t ) = 19 + 47 st . Output : The Newton polytope of the implicit equation g ( x , y , z ). STRATEGY: Recover the Newton polytope of g ( x , y , z ) from the Newton polytopes of the input polynomials f 1 , f 2 , f 3 . M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 3 / 10
x = f 1 ( s , t ) = 3 + 5 s + 7 t , y = f 2 ( s , t ) = 17 + 13 t + 11 s 2 , � Newton polytope of g ( x , y , z ). Y = z = f 3 ( s , t ) = 19 + 47 st . ρ 1 t st 2 P 1 P 3 ρ 2 1 ρ 6 s 1 t P 2 ρ 3 ρ 5 1 2 s 2 z 2 ρ 4 e 3 Γ � y 2 2 3 1 f -vector= (5, 8, 5) 2 e 1 e 2 2 ( − 1 , − 2 − 2) ( − 2 , − 2 − 3) x 2 y x 3 • Γ is a balanced weighted planar graph in R 3 . It is the tropical variety T ( g ( x , y , z )), dual to the Newton polytope of g . • We can recover g ( x , y , z ) from Γ using numerical linear algebra . M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 4 / 10
What is Tropical Geometry? Given a variety X ⊂ T n with defining ideal I ⊂ C [ x ± 1 1 , . . . , x ± 1 n ], the tropicalization of X equals: T X = T I := { w ∈ R n | in w I contains no monomial } . 1 It is a rational polyhedral fan in R n � T X ∩ S n − 1 is a spherical polyhedral complex. 2 If I is prime, then T X is pure of the same dimension as X . 3 Maximal cones have canonical multiplicities attached to them. Example (hypersurfaces): Maximal cones in T ( g ) are dual to edges in the Newton polytope NP ( g ), and m σ is the lattice length of the associated edge. Multiplicities are essential to recover NP ( g ) from T ( g ). M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 5 / 10
What is Geometric Tropicalization? AIM: Given Z ⊂ T N a surface , compute T Z from the geometry of Z . KEY FACT: T Z can be characterized in terms of divisorial valuations. M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 6 / 10
What is Geometric Tropicalization? AIM: Given Z ⊂ T N a surface , compute T Z from the geometry of Z . KEY FACT: T Z can be characterized in terms of divisorial valuations. Theorem (Geometric Tropicalization [Hacking - Keel - Tevelev, C . ]) Consider T N with coordinate functions χ 1 , . . . , χ N , and let Z ⊂ T N be a closed smooth surface . Suppose Z ⊃ Z is any normal and Q -factorial compactification, whose boundary divisor has m irreducible components D 1 , . . . , D m with no triple intersections ( C.N.C. ). Let ∆ be the graph: V (∆) = { 1 , . . . , m } ; ( i , j ) ∈ E (∆) ⇐ ⇒ D i ∩ D j � = ∅ . Realize ∆ as a graph Γ ⊂ R N by [ D k ]:=( val D k ( χ 1 ) , . . . , val D k ( χ N )) ∈ Z N . Then, T Z is the cone over the graph Γ . M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 6 / 10
What is Geometric Tropicalization? AIM: Given Z ⊂ T N a surface , compute T Z from the geometry of Z . KEY FACT: T Z can be characterized in terms of divisorial valuations. Theorem (Geometric Tropicalization [Hacking - Keel - Tevelev, C . ]) Consider T N with coordinate functions χ 1 , . . . , χ N , and let Z ⊂ T N be a closed smooth surface . Suppose Z ⊃ Z is any normal and Q -factorial compactification, whose boundary divisor has m irreducible components D 1 , . . . , D m with no triple intersections ( C.N.C. ). Let ∆ be the graph: V (∆) = { 1 , . . . , m } ; ( i , j ) ∈ E (∆) ⇐ ⇒ D i ∩ D j � = ∅ . Realize ∆ as a graph Γ ⊂ R N by [ D k ]:=( val D k ( χ 1 ) , . . . , val D k ( χ N )) ∈ Z N . Then, T Z is the cone over the graph Γ . Theorem (Combinatorial formula for multiplicities [C.]) � � ( Z � [ D i ] , [ D j ] � ) sat : Z � [ D i ] , [ D j ] � m ([ D i ] , [ D j ]) = ( D i · D j ) M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 6 / 10
QUESTION: How to compute T Y from a parameterization f = ( f 1 , . . . , f n ): T 2 ��� Y ⊂ T n ? M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 7 / 10
QUESTION: How to compute T Y from a parameterization f = ( f 1 , . . . , f n ): T 2 ��� Y ⊂ T n ? � n ANSWER: Compactify the domain X = T 2 � ( f i = 0) and use the i =1 map f to translate back to Y . Proposition Given f : X ⊂ T 2 → Y ⊂ T n generically finite map of degree δ , let X be a normal, Q -factorial, CNC compactification with intersection complex ∆ . Map each vertex D k of ∆ in Z n to a vertex [ � D k ] of Γ ⊂ R n , where [ � D k ] = val D k ( χ ◦ f ) = f # ([ D k ]) . Then, T Y is the cone over the graph Γ ⊂ R n , with multiplicities � � D j ]) = 1 D j ]) � sat : Z � [ � ( Z � [ � D i ] , [ � D i ] , [ � δ ( D i · D j ) D j ] � m ([ f . D i ] , [ f M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 7 / 10
Implicitization of generic surfaces SETTING: Let f = ( f 1 , . . . , f n ): T 2 ��� Y ⊂ T n of deg( f ) = δ , where we fix the Newton polytope of each f i and allow generic coefficients . GOAL: Compute the graph Γ of T Y from the Newton polytopes {P i } n i =1 . IDEA: Compactify X inside the proj. toric variety X N , where N is the common refinement of all N ( P i ). Generically , X is smooth with CNC. The vertices and edges of the boundary intersection complex ∆ are � { D ρ : ρ ∈ N [1] } , V (∆) = { E i : dim P i � = 0 , 1 ≤ i ≤ n } ( D ρ , D ρ ′ ) ∈ E (∆) iff ρ, ρ ′ are consecutive rays in N . ( E i , D ρ ) ∈ E (∆) iff ρ ∈ N ( P i ). ( E i , E j ) ∈ E (∆) iff ( f i = f j = 0) has a solution in T 2 . Then, Γ is the realization of ∆ via (1 ≤ i ≤ n ) , ∀ ray ρ ( η ρ prim. vector.) [ E i ] := e i [ D ρ ] := trop( f )( η ρ ) Theorem [Sturmfels-Tevelev-Yu, C.]: T Y is the weighted cone over Γ. M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 8 / 10
Implicitization of non-generic surfaces Non-genericity ↔ CNC condition is violated. 1 Embed X in X N . Solution 1: 2 Resolve triple intersections and singularities by classical blow-ups, and carry divisorial valuations along the way. M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 9 / 10
Implicitization of non-generic surfaces Non-genericity ↔ CNC condition is violated. 1 Embed X in X N . Solution 1: 2 Resolve triple intersections and singularities by classical blow-ups, and carry divisorial valuations along the way. 1 Embed X in P 2 Solution 2: ( s , t , u ) � n + 1 boundary divisors (1 ≤ i ≤ n ) , E i = ( f i = 0) E ∞ = ( u = 0) . 2 Resolve triple intersections and singularities by blow-ups π : ˜ X → X , and read divisorial valuations by columns � r ( f ◦ π ) ∗ ( χ i ) = π ∗ ( E i − deg( f i ) E ∞ ) = E ′ i − deg( f i ) E ′ ∞ − ∀ i . b ij H j j =1 The graph ∆ is obtained by gluing resolution diagrams and adding pairwise intersections. M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 9 / 10
Example (non-generic surface) z 2 z x = f 1 ( s , t ) = s − t , � y 2 y 1 (7 , 11 , 6) y = f 2 ( s , t ) = t − s 2 , Y = x z = f 3 ( s , t ) = − 1 + s t , x 2 y e 3 Affine Charts: x 3 Γ E 3 E 3 E 3 E 2 E 2 E 2 E 2 2 ( − 1 , − 1 − 1) � e 1 e 2 2 E 1 := ( s − t = 0) E 1 := ( s − t = 0) E 1 := ( s − t = 0) E 1 := ( s − t = 0) E 1 := ( s − t = 0) E 2 := ( t − s 2 = 0) E 2 := ( t − s 2 = 0) E 2 := ( t − s 2 = 0) E 2 := ( t − s 2 = 0) ( − 1 , − 2 − 2) ( − 2 , − 2 − 3) E 3 := (1 − st = 0) E 3 := (1 − st = 0) E 3 := (1 − st = 0) E 1 E 1 E 1 E 1 E 1 E 3 E 3 E 3 � E 2 E 2 E 2 E 2 E 2 E ∞ E ∞ � E 2 := ( u − s 2 = 0) E 2 := ( u − s 2 = 0) E 2 := ( u − s 2 = 0) E 2 := ( u − s 2 = 0) E 2 := ( u − s 2 = 0) E 3 := ( − u 2 + s = 0) E 3 := ( − u 2 + s = 0) E 3 := ( − u 2 + s = 0) E ∞ := ( u = 0) E ∞ := ( u = 0) M.A. Cueto (Columbia Univ.) Tropical Implicitization of surfaces Oct. 7 2011 10 / 10
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