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 = jV (P)j − 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 = jV (P)j − 3) AP Generalization to higher-dimensional polytopes? I - IX II If P is a polygon, then Z(adjP ) = AP∗ . ∗ n (Recall: P = fx 2 R j 8v 2 V (P): `v (x) ≥ 0g dual polytope of P) X Y Definition adjτ(P)(t) := vol(σ) `v (t); σ2τ(P) v2V (P)nV (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 = fx 2 R j 8v 2 V (P): `v (x) ≥ 0g 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); σ2τ(P) v2V (P)nV (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 = fx 2 R j 8v 2 V (P): `v (x) ≥ 0g 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); σ2τ(P) v2V (P)nV (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); σ2τ(P) v2V (P)nV (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 = fx 2 R j 8v 2 V (P): `v (x) ≥ 0g 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); σ2τ(P) v2V (P)nV (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 = fx 2 R j 8v 2 V (P): `v (x) ≥ 0g 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.