The Adjoint of a Polytope

joint works with Kristian Ranestad (Universitetet i Oslo) / Boris Shapiro (Stockholms universitet) & Bernd Sturmfels (MPI MiS Leipzig, UC Berkeley)

April 14, 2019 Generalization to higher-dimensional polytopes?

The Adjoint of a Polygon Wachspress (1975)

Definition 2 The adjoint AP of a polygon P ⊂ P is the minimal degree curve passing through the intersection points of pairs of lines containing non-adjacent edges of P.

(deg AP = |V (P)| − 3)

AP

I - IX The Adjoint of a Polygon Wachspress (1975)

Definition 2 The adjoint AP of a polygon P ⊂ P is the minimal degree curve passing through the intersection points of pairs of lines containing non-adjacent edges of P.

(deg AP = |V (P)| − 3)

AP

Generalization to higher-dimensional polytopes?

I - IX II If P is a polygon, then Z(adjP ) = AP∗ . ∗ n (Recall: P = {x ∈ R | ∀v ∈ V (P): `v (x) ≥ 0} dual polytope of P)

X Y Definition adjτ(P)(t) := vol(σ) `v (t), σ∈τ(P) v∈V (P)\V (σ)

where t = (t1,..., tn) and `v (t) = 1 − v1t1 − v2t2 − ... − vntn.

Theorem (Warren)

I adjτ(P)(t) is independent of the triangulation τ(P). So adjP := adjτ(P).

Geometric definition using a vanishing condition `ala Wachspress?

The Adjoint of a Polytope Warren (1996) n P: convex polytope in R V (P): set of vertices of P τ(P): triangulation of P using only the vertices of P

II - IX II If P is a polygon, then Z(adjP ) = AP∗ . ∗ n (Recall: P = {x ∈ R | ∀v ∈ V (P): `v (x) ≥ 0} dual polytope of P)

Theorem (Warren)

I adjτ(P)(t) is independent of the triangulation τ(P). So adjP := adjτ(P).

Geometric definition using a vanishing condition `ala Wachspress?

The Adjoint of a Polytope Warren (1996) n P: convex polytope in R V (P): set of vertices of P τ(P): triangulation of P using only the vertices of P X Y Definition adjτ(P)(t) := vol(σ) `v (t), σ∈τ(P) v∈V (P)\V (σ) where t = (t1,..., tn) and `v (t) = 1 − v1t1 − v2t2 − ... − vntn.

II - IX II If P is a polygon, then Z(adjP ) = AP∗ . ∗ n (Recall: P = {x ∈ R | ∀v ∈ V (P): `v (x) ≥ 0} dual polytope of P)

Geometric definition using a vanishing condition `ala Wachspress?

The Adjoint of a Polytope Warren (1996) n P: convex polytope in R V (P): set of vertices of P τ(P): triangulation of P using only the vertices of P X Y Definition adjτ(P)(t) := vol(σ) `v (t), σ∈τ(P) v∈V (P)\V (σ) where t = (t1,..., tn) and `v (t) = 1 − v1t1 − v2t2 − ... − vntn.

Theorem (Warren)

I adjτ(P)(t) is independent of the triangulation τ(P). So adjP := adjτ(P).

II - IX Geometric definition using a vanishing condition `ala Wachspress?

The Adjoint of a Polytope Warren (1996) n P: convex polytope in R V (P): set of vertices of P τ(P): triangulation of P using only the vertices of P X Y Definition adjτ(P)(t) := vol(σ) `v (t), σ∈τ(P) v∈V (P)\V (σ) where t = (t1,..., tn) and `v (t) = 1 − v1t1 − v2t2 − ... − vntn.

Theorem (Warren)

I adjτ(P)(t) is independent of the triangulation τ(P). So adjP := adjτ(P). II If P is a polygon, then Z(adjP ) = AP∗ . ∗ n (Recall: P = {x ∈ R | ∀v ∈ V (P): `v (x) ≥ 0} dual polytope of P)

II - IX The Adjoint of a Polytope Warren (1996) n P: convex polytope in R V (P): set of vertices of P τ(P): triangulation of P using only the vertices of P X Y Definition adjτ(P)(t) := vol(σ) `v (t), σ∈τ(P) v∈V (P)\V (σ) where t = (t1,..., tn) and `v (t) = 1 − v1t1 − v2t2 − ... − vntn.

Theorem (Warren)

I adjτ(P)(t) is independent of the triangulation τ(P). So adjP := adjτ(P). II If P is a polygon, then Z(adjP ) = AP∗ . ∗ n (Recall: P = {x ∈ R | ∀v ∈ V (P): `v (x) ≥ 0} dual polytope of P)

Geometric definition using a vanishing condition `ala Wachspress? II - IX there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

with d facets

RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes),

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P

III - IX there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

with d facets

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes),

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

III - IX there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

with d facets

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes),

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

III - IX there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

with d facets

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes),

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

III - IX there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

with d facets

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes),

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

III - IX there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

with d facets

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes),

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

III - IX there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

with d facets

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes),

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

III - IX with d facets

adjoint double plane adjoint quadric surface adjoint plane

there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

The Adjoint of a Polytope n P: polytope in P HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), III - IX adjoint double plane adjoint quadric surface adjoint plane

there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

The Adjoint of a Polytope n P: polytope in P with d facets HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), III - IX adjoint double plane adjoint quadric surface adjoint plane

The Adjoint of a Polytope n P: polytope in P with d facets HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P. III - IX adjoint double plane adjoint quadric surface

The Adjoint of a Polytope n P: polytope in P with d facets HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P. III - IX adjoint double plane

The Adjoint of a Polytope n P: polytope in P with d facets HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P. III - IX The Adjoint of a Polytope n P: polytope in P with d facets HP : hyperplane arrangement spanned by facets of P RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

adjoint double plane adjoint quadric surface adjoint plane Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P. III - IX Proposition (K., Ranestad)

Warren’s adjoint polynomial adjP vanishes along RP∗ . If HP∗ is simple, then Z(adjP ) = AP∗ .

The Adjoint of a Polytope

n P: polytope in P with d facets HP : hyperplane arrangement spanned by facets of P

RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

IV - IX The Adjoint of a Polytope

n P: polytope in P with d facets HP : hyperplane arrangement spanned by facets of P

RP : residual arrangement of linear spaces that are intersections of hyperplanes in HP and do not contain any of face of P

Theorem (K., Ranestad) n If HP is simple (i.e. through any point in P pass ≤ n hyperplanes), there is n a unique hypersurface AP in P of degree d − n − 1 passing through RP . AP is called the adjoint of P.

Proposition (K., Ranestad)

Warren’s adjoint polynomial adjP vanishes along RP∗ . If HP∗ is simple, then Z(adjP ) = AP∗ . IV - IX Definition n Let P be a convex polytope in R . A set of ◦ functions {βu : P → R | u ∈ V (P)} is called generalized barycentric coordinates for P if, for all p ∈ P◦, Barycentric coordinates for simplices are uniquely (i) ∀u ∈ V (P): βu(p) > 0, P determined from (i)-(iii). (ii) βu(p) = 1, and u∈V (P) This is not true for other polytopes! P (iii) βu(p)u = p. u∈V (P)

Application 1: Barycentric Coordinates

area(4i ) βvi (p) := area(41) + area(42) + area(43) for i = 1, 2, 3

V - IX Barycentric coordinates for simplices are uniquely determined from (i)-(iii).

This is not true for other polytopes!

Application 1: Barycentric Coordinates

area(4i ) βvi (p) := area(41) + area(42) + area(43) for i = 1, 2, 3

Definition n Let P be a convex polytope in R . A set of ◦ functions {βu : P → R | u ∈ V (P)} is called generalized barycentric coordinates for P if, for all p ∈ P◦,

(i) ∀u ∈ V (P): βu(p) > 0, P (ii) βu(p) = 1, and u∈V (P) P (iii) βu(p)u = p. u∈V (P) V - IX Application 1: Barycentric Coordinates

area(4i ) βvi (p) := area(41) + area(42) + area(43) for i = 1, 2, 3

Definition n Let P be a convex polytope in R . A set of ◦ functions {βu : P → R | u ∈ V (P)} is called generalized barycentric coordinates for P if, for all p ∈ P◦, Barycentric coordinates for simplices are uniquely (i) ∀u ∈ V (P): βu(p) > 0, P determined from (i)-(iii). (ii) βu(p) = 1, and u∈V (P) This is not true for other polytopes! P (iii) βu(p)u = p. u∈V (P) V - IX V (P) ←→1:1 F(P∗) F(P) ←→1:1 V (P∗)

v 7−→ Fv F 7−→ vF

Proposition (Warren) For other GBCs and The Wachspress coordinates applications of GBCs (e.g., Q mesh parameterizations in adjFu (t) · `vF (t) F ∈F(P): u∈/F geometric modelling, βu(t) := deformations in computer adj ∗ (t) P graphics, or polyhedral FEM): for u ∈ V (P) [Floater: Generalized barycentric coordinates and applications, Acta are generalized barycentric coordinates for P. Numerica 24 (2015)]

Application 1: Barycentric Coordinates Warren (1996) n P: convex polytope in R F(P): set of facets of P

VI - IX F(P) ←→1:1 V (P∗)

F 7−→ vF

Proposition (Warren) For other GBCs and The Wachspress coordinates applications of GBCs (e.g., Q mesh parameterizations in adjFu (t) · `vF (t) F ∈F(P): u∈/F geometric modelling, βu(t) := deformations in computer adj ∗ (t) P graphics, or polyhedral FEM): for u ∈ V (P) [Floater: Generalized barycentric coordinates and applications, Acta are generalized barycentric coordinates for P. Numerica 24 (2015)]

Application 1: Barycentric Coordinates Warren (1996) n P: convex polytope in R F(P): set of facets of P V (P) ←→1:1 F(P∗)

v 7−→ Fv

VI - IX Proposition (Warren) For other GBCs and The Wachspress coordinates applications of GBCs (e.g., Q mesh parameterizations in adjFu (t) · `vF (t) F ∈F(P): u∈/F geometric modelling, βu(t) := deformations in computer adj ∗ (t) P graphics, or polyhedral FEM): for u ∈ V (P) [Floater: Generalized barycentric coordinates and applications, Acta are generalized barycentric coordinates for P. Numerica 24 (2015)]

Application 1: Barycentric Coordinates Warren (1996) n P: convex polytope in R F(P): set of facets of P V (P) ←→1:1 F(P∗) F(P) ←→1:1 V (P∗)

v 7−→ Fv F 7−→ vF

VI - IX For other GBCs and applications of GBCs (e.g., mesh parameterizations in geometric modelling, deformations in computer graphics, or polyhedral FEM): [Floater: Generalized barycentric coordinates and applications, Acta Numerica 24 (2015)]

Application 1: Barycentric Coordinates Warren (1996) n P: convex polytope in R F(P): set of facets of P V (P) ←→1:1 F(P∗) F(P) ←→1:1 V (P∗)

v 7−→ Fv F 7−→ vF

Proposition (Warren) The Wachspress coordinates Q adjFu (t) · `vF (t) F ∈F(P): u∈/F βu(t) := adjP∗ (t) for u ∈ V (P) are generalized barycentric coordinates for P. VI - IX Application 1: Barycentric Coordinates Warren (1996) n P: convex polytope in R F(P): set of facets of P V (P) ←→1:1 F(P∗) F(P) ←→1:1 V (P∗)

v 7−→ Fv F 7−→ vF

Proposition (Warren) For other GBCs and The Wachspress coordinates applications of GBCs (e.g., Q mesh parameterizations in adjFu (t) · `vF (t) F ∈F(P): u∈/F geometric modelling, βu(t) := deformations in computer adj ∗ (t) P graphics, or polyhedral FEM): for u ∈ V (P) [Floater: Generalized barycentric coordinates and applications, Acta are generalized barycentric coordinates for P. Numerica 24 (2015)] VI - IX moments Z i1 i2 in n mI (P) := w1 w2 ... wn dµP for I = (i1, i2,..., in) ∈ Z≥0 Rn

Proposition (K., Shapiro, Sturmfels)

X I adjP(t) cI mI (P) t = Q , vol(P) `v (t) I∈ n Z≥0 v∈V (P)

where c := i1+i2+...+in+n
. I i1,i2,...,in,n

Application 2: Moments of Probability Distributions K., Shapiro, Sturmfels n P: convex polytope in R µP : uniform probability distribution on P

VII - IX Proposition (K., Shapiro, Sturmfels)

X I adjP(t) cI mI (P) t = Q , vol(P) `v (t) I∈ n Z≥0 v∈V (P)

where c := i1+i2+...+in+n
. I i1,i2,...,in,n

Application 2: Moments of Probability Distributions K., Shapiro, Sturmfels n P: convex polytope in R µP : uniform probability distribution on P moments Z i1 i2 in n mI (P) := w1 w2 ... wn dµP for I = (i1, i2,..., in) ∈ Z≥0 Rn

VII - IX Application 2: Moments of Probability Distributions K., Shapiro, Sturmfels n P: convex polytope in R µP : uniform probability distribution on P moments Z i1 i2 in n mI (P) := w1 w2 ... wn dµP for I = (i1, i2,..., in) ∈ Z≥0 Rn

Proposition (K., Shapiro, Sturmfels)

X I adjP(t) cI mI (P) t = Q , vol(P) `v (t) I∈ n Z≥0 v∈V (P)

where c := i1+i2+...+in+n
. I i1,i2,...,in,n VII - IX c RP : codimension-c part of RP n P X smooth

D D˜ smooth polytopal hypersurface: hypersurface of degree d, c When can we find multiplicity c along RP , smooth outside of RP a polytopal hypersurface D? ˜ Adjunction formula: KD˜ = (KX + [D])|D˜ Def.: An adjoint to D˜ in X is a hypersurface A in X s.t. [A] = KX + [D˜]. Proposition (K., Ranestad) D˜ has a unique adjoint A in X , and thus a unique canonical divisor: A ∩ D˜. Moreover, π(A) = AP .

Why “Adjoint”? n P: polytope in P with d facets HP : simple hyperplane arrangement spanned by facets of P

blowup π Idea:

P HP

hypersurface of degree d

VIII - IX n P X smooth

D˜ smooth

When can we find a polytopal hypersurface D? ˜ Adjunction formula: KD˜ = (KX + [D])|D˜ Def.: An adjoint to D˜ in X is a hypersurface A in X s.t. [A] = KX + [D˜]. Proposition (K., Ranestad) D˜ has a unique adjoint A in X , and thus a unique canonical divisor: A ∩ D˜. Moreover, π(A) = AP .

Why “Adjoint”? n P: polytope in P with d facets HP : simple hyperplane arrangement spanned by facets of P c RP : codimension-c part of RP blowup π Idea:

P HP D polytopal hypersurface: hypersurface hypersurface of degree d, c of degree d multiplicity c along RP , smooth outside of RP

VIII - IX When can we find a polytopal hypersurface D? ˜ Adjunction formula: KD˜ = (KX + [D])|D˜ Def.: An adjoint to D˜ in X is a hypersurface A in X s.t. [A] = KX + [D˜]. Proposition (K., Ranestad) D˜ has a unique adjoint A in X , and thus a unique canonical divisor: A ∩ D˜. Moreover, π(A) = AP .

Why “Adjoint”? n P: polytope in P with d facets HP : simple hyperplane arrangement spanned by facets of P c RP : codimension-c part of RP n blowup π Idea: P X smooth

P HP D D˜ smooth polytopal hypersurface: hypersurface hypersurface of degree d, c of degree d multiplicity c along RP , smooth outside of RP

VIII - IX When can we find a polytopal hypersurface D?

Def.: An adjoint to D˜ in X is a hypersurface A in X s.t. [A] = KX + [D˜]. Proposition (K., Ranestad) D˜ has a unique adjoint A in X , and thus a unique canonical divisor: A ∩ D˜. Moreover, π(A) = AP .

Why “Adjoint”? n P: polytope in P with d facets HP : simple hyperplane arrangement spanned by facets of P c RP : codimension-c part of RP n blowup π Idea: P X smooth

P HP D D˜ smooth polytopal hypersurface: hypersurface hypersurface of degree d, c of degree d multiplicity c along RP , smooth outside of RP ˜ Adjunction formula: KD˜ = (KX + [D])|D˜

VIII - IX When can we find a polytopal hypersurface D?

Proposition (K., Ranestad) D˜ has a unique adjoint A in X , and thus a unique canonical divisor: A ∩ D˜. Moreover, π(A) = AP .

Why “Adjoint”? n P: polytope in P with d facets HP : simple hyperplane arrangement spanned by facets of P c RP : codimension-c part of RP n blowup π Idea: P X smooth

P HP D D˜ smooth polytopal hypersurface: hypersurface hypersurface of degree d, c of degree d multiplicity c along RP , smooth outside of RP ˜ Adjunction formula: KD˜ = (KX + [D])|D˜ Def.: An adjoint to D˜ in X is a hypersurface A in X s.t. [A] = KX + [D˜].

VIII - IX When can we find a polytopal hypersurface D?

Why “Adjoint”? n P: polytope in P with d facets HP : simple hyperplane arrangement spanned by facets of P c RP : codimension-c part of RP n blowup π Idea: P X smooth

P HP D D˜ smooth polytopal hypersurface: hypersurface hypersurface of degree d, c of degree d multiplicity c along RP , smooth outside of RP ˜ Adjunction formula: KD˜ = (KX + [D])|D˜ Def.: An adjoint to D˜ in X is a hypersurface A in X s.t. [A] = KX + [D˜]. Proposition (K., Ranestad) D˜ has a unique adjoint A in X , and thus a unique canonical divisor: A ∩ D˜. Moreover, π(A) = AP . VIII - IX Why “Adjoint”? n P: polytope in P with d facets HP : simple hyperplane arrangement spanned by facets of P c RP : codimension-c part of RP n blowup π Idea: P X smooth

P HP D D˜ smooth polytopal hypersurface: hypersurface hypersurface of degree d, c When can we find of degree d multiplicity c along RP , smooth outside of RP a polytopal hypersurface D? ˜ Adjunction formula: KD˜ = (KX + [D])|D˜ Def.: An adjoint to D˜ in X is a hypersurface A in X s.t. [A] = KX + [D˜]. Proposition (K., Ranestad) D˜ has a unique adjoint A in X , and thus a unique canonical divisor: A ∩ D˜. Moreover, π(A) = AP . VIII - IX Theorem (K., Ranestad) 3 Let C be a combinatorial type of simple polytopes in P and let P be a general polytope of type C. There is a polytopal surface D iff C is one of:

In that case, the general D is either an elliptic surface or a K3-surface.

Polytopal Hypersurfaces Proposition (K., Ranestad) 2 Let P be a general d-gon in P . There is a polygonal curve D iff d ≤ 6. In that case, D is an elliptic curve.

IX - IX Polytopal Hypersurfaces Proposition (K., Ranestad) 2 Let P be a general d-gon in P . There is a polygonal curve D iff d ≤ 6. In that case, D is an elliptic curve. Theorem (K., Ranestad) 3 Let C be a combinatorial type of simple polytopes in P and let P be a general polytope of type C. There is a polytopal surface D iff C is one of:

In that case, the general D is either an elliptic surface or a K3-surface. IX - IX