Slack matrices of polytopes Let P be a polytope with facets given by h 1 ( x ) ≥ 0 , . . . , h f ( x ) ≥ 0, and vertices p 1 , . . . , p v . The slack matrix of P is the matrix S P ∈ R v × f given by S P ( i , j ) = h j ( p i ) . Its 6 × 6 slack matrix. Regular hexagon. 0 0 1 2 2 1 1 0 0 1 2 2 2 1 0 0 1 2 2 2 1 0 0 1 1 2 2 1 0 0 0 1 2 2 1 0 The slack matrix is defined only up to column scaling; João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 9 / 27
Slack matrices of polytopes Let P be a polytope with facets given by h 1 ( x ) ≥ 0 , . . . , h f ( x ) ≥ 0, and vertices p 1 , . . . , p v . The slack matrix of P is the matrix S P ∈ R v × f given by S P ( i , j ) = h j ( p i ) . Its 6 × 6 slack matrix. Regular hexagon. 0 0 1 2 2 1 1 0 0 1 2 2 2 1 0 0 1 2 2 2 1 0 0 1 1 2 2 1 0 0 0 1 2 2 1 0 The slack matrix is defined only up to column scaling; The slack matrix can’t see affine transformations; Moreover P is affinely equivalent to the convex hull of the rows of S P . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 9 / 27
Characterization of slack matrices If P is a d -polytope with V -representation { p 1 , . . . , p v } and H -representation Ax ≤ b then − A � � � 1 1 · · · 1 � b S P = p 1 p 2 · · · p v In particular S P has rank d + 1. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 10 / 27
Characterization of slack matrices If P is a d -polytope with V -representation { p 1 , . . . , p v } and H -representation Ax ≤ b then − A � � � 1 1 · · · 1 � b S P = p 1 p 2 · · · p v In particular S P has rank d + 1. Any polytope of the same combinatorial class of P must have a slack matrix with the same zero-pattern. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 10 / 27
Characterization of slack matrices If P is a d -polytope with V -representation { p 1 , . . . , p v } and H -representation Ax ≤ b then − A � � � 1 1 · · · 1 � b S P = p 1 p 2 · · · p v In particular S P has rank d + 1. Any polytope of the same combinatorial class of P must have a slack matrix with the same zero-pattern. Theorem (GGKPRT, 2013) A nonnegative matrix S is the slack matrix of some realization of P if and only if supp ( S ) = supp ( S P ) ; 1 rank ( S ) = rank ( S P ) = d + 1 ; 2 the all ones vector lies in the column span of S. 3 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 10 / 27
Characterization of slack matrices If P is a d -polytope with V -representation { p 1 , . . . , p v } and H -representation Ax ≤ b then − A � � � 1 1 · · · 1 � b S P = p 1 p 2 · · · p v In particular S P has rank d + 1. Any polytope of the same combinatorial class of P must have a slack matrix with the same zero-pattern. Theorem (GGKPRT, 2013) A nonnegative matrix S is the slack matrix of some realization of P if and only if supp ( S ) = supp ( S P ) ; 1 rank ( S ) = rank ( S P ) = d + 1 ; 2 the all ones vector lies in the column span of S. 3 There is a one-to-one correspondence between matrices with those properties (up to column scaling) and realizations of P (up to affine equivalence). João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 10 / 27
Projective equivalence In general, we will be interested in modding out projective transformations. = P ⇔ Q = φ ( P ) , φ ( x ) = Ax + b � � A b p c ⊺ + d , det � = 0 Q c ⊺ x d João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 11 / 27
Projective equivalence In general, we will be interested in modding out projective transformations. = P ⇔ Q = φ ( P ) , φ ( x ) = Ax + b � � A b p c ⊺ + d , det � = 0 Q c ⊺ x d All convex quadrilaterals are projectively equivalent to a square. A square is projectively unique. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 11 / 27
Projective equivalence In general, we will be interested in modding out projective transformations. = P ⇔ Q = φ ( P ) , φ ( x ) = Ax + b � � A b p c ⊺ + d , det � = 0 Q c ⊺ x d All convex quadrilaterals are projectively equivalent to a square. A square is projectively unique. Slack matrices offer a natural way of quotient projective transformations. Theorem (GPRT, 2017) p Q = P ⇔ S Q = D v S P D f for some positive diagonal matrices D v , D f João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 11 / 27
Slack ideals Slack ideal Let P be a d -polytope and S P ( x ) a symbolic matrix with the same support as S P . Then the slack ideal of P is I P = � ( d + 2 ) -minors of S P ( x ) � . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 12 / 27
Slack ideals Slack ideal Let P be a d -polytope and S P ( x ) a symbolic matrix with the same support as S P . Then the slack ideal of P is � x i ) ∞ . I P = � ( d + 2 ) -minors of S P ( x ) � : ( João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 12 / 27
Slack ideals Slack ideal Let P be a d -polytope and S P ( x ) a symbolic matrix with the same support as S P . Then the slack ideal of P is � x i ) ∞ . I P = � ( d + 2 ) -minors of S P ( x ) � : ( João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 12 / 27
Slack ideals Slack ideal Let P be a d -polytope and S P ( x ) a symbolic matrix with the same support as S P . Then the slack ideal of P is � x i ) ∞ . I P = � ( d + 2 ) -minors of S P ( x ) � : ( 1 1 0 0 0 0 1 1 0 0 S P = 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 12 / 27
Slack ideals Slack ideal Let P be a d -polytope and S P ( x ) a symbolic matrix with the same support as S P . Then the slack ideal of P is � x i ) ∞ . I P = � ( d + 2 ) -minors of S P ( x ) � : ( 1 1 0 0 0 x 1 x 2 0 0 0 0 1 1 0 0 0 0 0 x 3 x 4 S P = 0 0 1 1 0 S P ( x ) = 0 0 x 5 x 6 0 1 0 0 1 0 x 7 0 0 x 8 0 0 0 0 0 1 0 0 0 0 x 9 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 12 / 27
Slack ideals Slack ideal Let P be a d -polytope and S P ( x ) a symbolic matrix with the same support as S P . Then the slack ideal of P is � x i ) ∞ . I P = � ( d + 2 ) -minors of S P ( x ) � : ( 1 1 0 0 0 x 1 x 2 0 0 0 0 1 1 0 0 0 0 0 x 3 x 4 S P = 0 0 1 1 0 S P ( x ) = 0 0 x 5 x 6 0 1 0 0 1 0 x 7 0 0 x 8 0 0 0 0 0 1 0 0 0 0 x 9 � I P = � x 1 x 3 x 5 x 8 x 9 − x 2 x 4 x 6 x 7 x 9 � : ( x i ) ∞ João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 12 / 27
Slack ideals Slack ideal Let P be a d -polytope and S P ( x ) a symbolic matrix with the same support as S P . Then the slack ideal of P is � x i ) ∞ . I P = � ( d + 2 ) -minors of S P ( x ) � : ( 1 1 0 0 0 x 1 x 2 0 0 0 0 1 1 0 0 0 0 0 x 3 x 4 S P = 0 0 1 1 0 S P ( x ) = 0 0 x 5 x 6 0 1 0 0 1 0 x 7 0 0 x 8 0 0 0 0 0 1 0 0 0 0 x 9 x i ) ∞ = � x 1 x 3 x 5 x 8 − x 2 x 4 x 6 x 7 � � I P = � x 1 x 3 x 5 x 8 x 9 − x 2 x 4 x 6 x 7 x 9 � : ( João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 12 / 27
Slack realization space V ( I P ) is the slack variety of P . Positive part of slack variety: V + ( I P ) = V ( I P ) ∩ R n + João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 13 / 27
Slack realization space V ( I P ) is the slack variety of P . Positive part of slack variety: V + ( I P ) = V ( I P ) ∩ R n + > 0 × R f R v > 0 acts on V + ( I P ) : for every s ∈ V + ( I P ) , D v s D f ∈ V + ( I P ) D v , D f positive diagonal matrices João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 13 / 27
Slack realization space V ( I P ) is the slack variety of P . Positive part of slack variety: V + ( I P ) = V ( I P ) ∩ R n + > 0 × R f R v > 0 acts on V + ( I P ) : for every s ∈ V + ( I P ) , D v s D f ∈ V + ( I P ) D v , D f positive diagonal matrices Theorem (GMTW, 2017) 1 : 1 > 0 × R f V + ( I P ) / ( R v > 0 ) ← → classes of projectively equivalent polytopes of the same combinatorial type as P. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 13 / 27
Slack realization space V ( I P ) is the slack variety of P . Positive part of slack variety: V + ( I P ) = V ( I P ) ∩ R n + > 0 × R f R v > 0 acts on V + ( I P ) : for every s ∈ V + ( I P ) , D v s D f ∈ V + ( I P ) D v , D f positive diagonal matrices Theorem (GMTW, 2017) 1 : 1 > 0 × R f V + ( I P ) / ( R v > 0 ) ← → classes of projectively equivalent polytopes of the same combinatorial type as P. > 0 × R f We call V + ( I P ) / ( R v > 0 ) the slack realization space of P . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 13 / 27
Connection to the classical model � p 1 � x = · · · p v ∈ R ( P ) João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 14 / 27
Connection to the classical model � 1 � · · · 1 � p 1 � x = · · · p v ∈ R ( P ) → x = p 1 · · · p v João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 14 / 27
Connection to the classical model � 1 � · · · 1 � p 1 � x = · · · p v ∈ R ( P ) → x = p 1 · · · p v ↓ row space of x ∈ Gr d + 1 ( R v ) João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 14 / 27
Connection to the classical model � 1 � · · · 1 � p 1 � x = · · · p v ∈ R ( P ) → x = p 1 · · · p v ↓ x = (det( x I )) I ∈ P ( d ) − 1 v row space of x ∈ Gr d + 1 ( R v ) ˜ ← João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 14 / 27
Connection to the classical model � 1 � · · · 1 � p 1 � x = · · · p v ∈ R ( P ) → x = p 1 · · · p v ↓ x = (det( x I )) I ∈ P ( d ) − 1 v row space of x ∈ Gr d + 1 ( R v ) ˜ ← This sends R ( P ) bijectively up to affine transformations into a subset of the Plücker embedding of Gr d + 1 ( R v ) cut out (mostly) from positivity, negativity and nullity conditions on some of the variables. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 14 / 27
Connection to the classical model � 1 � · · · 1 � p 1 � x = · · · p v ∈ R ( P ) → x = p 1 · · · p v ↓ x = (det( x I )) I ∈ P ( d ) − 1 v row space of x ∈ Gr d + 1 ( R v ) ˜ ← This sends R ( P ) bijectively up to affine transformations into a subset of the Plücker embedding of Gr d + 1 ( R v ) cut out (mostly) from positivity, negativity and nullity conditions on some of the variables. If for every facet k of P we pick a set I k of d − 1 spanning vertices we can define a matrix ( S (˜ x )) k , l = ± ˜ x ( I k , l ) This is a slack matrix of P and its row space is ¯ x . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 14 / 27
Section 3 Applications João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 15 / 27
Application 1: Psd-minimality A semidefinite representation of size k of a d -polytope P is a description x ∈ R d � � � � � P = � ∃ y s.t. A 0 + A i x i + B i y i � 0 � where A i and B i are k × k real symmetric matrices. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 16 / 27
Application 1: Psd-minimality A semidefinite representation of size k of a d -polytope P is a description x ∈ R d � � � � � P = � ∃ y s.t. A 0 + A i x i + B i y i � 0 � where A i and B i are k × k real symmetric matrices. If we allow A i and B i to be hermitian, we call it a complex semidefinite representation. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 16 / 27
Application 1: Psd-minimality A semidefinite representation of size k of a d -polytope P is a description x ∈ R d � � � � � P = � ∃ y s.t. A 0 + A i x i + B i y i � 0 � where A i and B i are k × k real symmetric matrices. If we allow A i and B i to be hermitian, we call it a complex semidefinite representation. Projection on x 1 and x 2 of 1 x 1 x 2 � 0 . x 1 x 1 y x 2 y x 2 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 16 / 27
Application 1: Psd-minimality A semidefinite representation of size k of a d -polytope P is a description x ∈ R d � � � � � P = � ∃ y s.t. A 0 + A i x i + B i y i � 0 � where A i and B i are k × k real symmetric matrices. If we allow A i and B i to be hermitian, we call it a complex semidefinite representation. Projection on x 1 and x 2 of 1 x 1 x 2 � 0 . x 1 x 1 y x 2 y x 2 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 16 / 27
Application 1: Psd-minimality A semidefinite representation of size k of a d -polytope P is a description x ∈ R d � � � � � P = � ∃ y s.t. A 0 + A i x i + B i y i � 0 � where A i and B i are k × k real symmetric matrices. If we allow A i and B i to be hermitian, we call it a complex semidefinite representation. Projection on x 1 and x 2 of 1 x 1 x 2 � 0 . x 1 x 1 y x 2 y x 2 Optimizing over such sets is “easy”: we want small representations. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 16 / 27
Application 1: Psd-minimality A semidefinite representation of size k of a d -polytope P is a description x ∈ R d � � � � � P = � ∃ y s.t. A 0 + A i x i + B i y i � 0 � where A i and B i are k × k real symmetric matrices. If we allow A i and B i to be hermitian, we call it a complex semidefinite representation. Projection on x 1 and x 2 of 1 x 1 x 2 � 0 . x 1 x 1 y x 2 y x 2 Optimizing over such sets is “easy”: we want small representations. Turns out the smallest possible size is d + 1. When does that happen? João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 16 / 27
Application 1: Psd-minimality (part 2) Theorem (GRT 2013; GGS 2016) A polytope P is psd-minimal ⇔ ∃ S p ( y ) ∈ V R ( I P ) such that S P = S P ( y 2 ) . A polytope P is psd C -minimal ⇔ ∃ S p ( y ) ∈ V C ( I P ) such that S P = S P ( | y | 2 ) João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 17 / 27
Application 1: Psd-minimality (part 2) Theorem (GRT 2013; GGS 2016) A polytope P is psd-minimal ⇔ ∃ S p ( y ) ∈ V R ( I P ) such that S P = S P ( y 2 ) . A polytope P is psd C -minimal ⇔ ∃ S p ( y ) ∈ V C ( I P ) such that S P = S P ( | y | 2 ) Lemma If I P has a trinomial x a + x b − x c then P is not psd-minimal. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 17 / 27
Application 1: Psd-minimality (part 2) Theorem (GRT 2013; GGS 2016) A polytope P is psd-minimal ⇔ ∃ S p ( y ) ∈ V R ( I P ) such that S P = S P ( y 2 ) . A polytope P is psd C -minimal ⇔ ∃ S p ( y ) ∈ V C ( I P ) such that S P = S P ( | y | 2 ) Lemma If I P has a trinomial x a + x b − x c then P is not psd-minimal. In R 2 (2 types), R 3 (6 types) this recovers [GRT 2013]. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 17 / 27
Application 1: Psd-minimality (part 2) Theorem (GRT 2013; GGS 2016) A polytope P is psd-minimal ⇔ ∃ S p ( y ) ∈ V R ( I P ) such that S P = S P ( y 2 ) . A polytope P is psd C -minimal ⇔ ∃ S p ( y ) ∈ V C ( I P ) such that S P = S P ( | y | 2 ) Lemma If I P has a trinomial x a + x b − x c then P is not psd-minimal. In R 2 (2 types), R 3 (6 types) this recovers [GRT 2013]. In R 4 (31 types) this allowed the classification [GPRT, 2017]. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 17 / 27
Application 1: Psd-minimality (part 2) Theorem (GRT 2013; GGS 2016) A polytope P is psd-minimal ⇔ ∃ S p ( y ) ∈ V R ( I P ) such that S P = S P ( y 2 ) . A polytope P is psd C -minimal ⇔ ∃ S p ( y ) ∈ V C ( I P ) such that S P = S P ( | y | 2 ) Lemma If I P has a trinomial x a + x b − x c then P is not psd-minimal. In R 2 (2 types), R 3 (6 types) this recovers [GRT 2013]. In R 4 (31 types) this allowed the classification [GPRT, 2017]. Lemma Suppose P is psd C -minimal, i.e. S P = S P ( | y | 2 ) . If I P has a trinomial x a + x b − x c then ℜ ( y a y b ) = 0 . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 17 / 27
Application 1: Psd-minimality (part 2) Theorem (GRT 2013; GGS 2016) A polytope P is psd-minimal ⇔ ∃ S p ( y ) ∈ V R ( I P ) such that S P = S P ( y 2 ) . A polytope P is psd C -minimal ⇔ ∃ S p ( y ) ∈ V C ( I P ) such that S P = S P ( | y | 2 ) Lemma If I P has a trinomial x a + x b − x c then P is not psd-minimal. In R 2 (2 types), R 3 (6 types) this recovers [GRT 2013]. In R 4 (31 types) this allowed the classification [GPRT, 2017]. Lemma Suppose P is psd C -minimal, i.e. S P = S P ( | y | 2 ) . If I P has a trinomial x a + x b − x c then ℜ ( y a y b ) = 0 . In R 2 (3 types), [GGS 2017, CG 2018]. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 17 / 27
Application 1: Psd-minimality (part 2) Theorem (GRT 2013; GGS 2016) A polytope P is psd-minimal ⇔ ∃ S p ( y ) ∈ V R ( I P ) such that S P = S P ( y 2 ) . A polytope P is psd C -minimal ⇔ ∃ S p ( y ) ∈ V C ( I P ) such that S P = S P ( | y | 2 ) Lemma If I P has a trinomial x a + x b − x c then P is not psd-minimal. In R 2 (2 types), R 3 (6 types) this recovers [GRT 2013]. In R 4 (31 types) this allowed the classification [GPRT, 2017]. Lemma Suppose P is psd C -minimal, i.e. S P = S P ( | y | 2 ) . If I P has a trinomial x a + x b − x c then ℜ ( y a y b ) = 0 . In R 2 (3 types), [GGS 2017, CG 2018]. In R 3 who knows?... João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 17 / 27
Application 2: Rationality A combinatorial polytope is rational if it has a realization in which all vertices have rational coordinates. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 18 / 27
Application 2: Rationality A combinatorial polytope is rational if it has a realization in which all vertices have rational coordinates. Lemma A polytope P is rational ⇔ V + ( I P ) has a rational point. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 18 / 27
Application 2: Rationality A combinatorial polytope is rational if it has a realization in which all vertices have rational coordinates. Lemma A polytope P is rational ⇔ V + ( I P ) has a rational point. We consider the following point-line arrangement in the plane [Grünbaum, 1967]: x 1 0 x 2 0 x 3 x 4 x 5 x 6 0 x 7 x 8 x 9 0 x 10 0 0 x 11 x 12 x 13 x 14 0 x 15 x 16 x 17 x 18 0 0 x 19 x 20 0 x 21 0 0 x 22 x 23 x 24 S P ( x ) = x 25 0 x 26 x 27 0 x 28 0 0 x 29 0 0 x 30 x 31 x 32 0 x 33 x 34 x 35 0 x 36 0 x 37 x 38 x 39 0 x 40 x 41 0 x 42 x 43 0 x 44 x 45 x 46 0 x 47 0 x 48 x 49 x 50 0 x 51 x 52 x 53 0 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 18 / 27
Application 2: Rationality A combinatorial polytope is rational if it has a realization in which all vertices have rational coordinates. Lemma A polytope P is rational ⇔ V + ( I P ) has a rational point. We consider the following point-line arrangement in the plane [Grünbaum, 1967]: x 1 0 x 2 0 x 3 x 4 x 5 x 6 0 x 7 x 8 x 9 0 x 10 0 0 x 11 x 12 x 13 x 14 0 x 15 x 16 x 17 x 18 0 0 x 19 x 20 0 x 21 0 0 x 22 x 23 x 24 S P ( x ) = x 25 0 x 26 x 27 0 x 28 0 0 x 29 0 0 x 30 x 31 x 32 0 x 33 x 34 x 35 0 x 36 0 x 37 x 38 x 39 0 x 40 x 41 0 x 42 x 43 0 x 44 x 45 x 46 0 x 47 0 x 48 x 49 x 50 0 x 51 x 52 x 53 0 Scaling rows and columns to set some variables to 1 (this does not affect rationality): x 2 46 + x 46 − 1 ∈ I P João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 18 / 27
Application 2: Rationality A combinatorial polytope is rational if it has a realization in which all vertices have rational coordinates. Lemma A polytope P is rational ⇔ V + ( I P ) has a rational point. We consider the following point-line arrangement in the plane [Grünbaum, 1967]: x 1 0 x 2 0 x 3 x 4 x 5 x 6 0 x 7 x 8 x 9 0 x 10 0 0 x 11 x 12 x 13 x 14 0 x 15 x 16 x 17 x 18 0 0 x 19 x 20 0 x 21 0 0 x 22 x 23 x 24 S P ( x ) = x 25 0 x 26 x 27 0 x 28 0 0 x 29 0 0 x 30 x 31 x 32 0 x 33 x 34 x 35 0 x 36 0 x 37 x 38 x 39 0 x 40 x 41 0 x 42 x 43 0 x 44 x 45 x 46 0 x 47 0 x 48 x 49 x 50 0 x 51 x 52 x 53 0 Scaling rows and columns to set some variables to 1 (this does not affect rationality): √ 46 + x 46 − 1 ∈ I P ⇒ x 46 = − 1 ± 5 x 2 ⇒ no rational realizations 2 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 18 / 27
Application 2: Rationality A combinatorial polytope is rational if it has a realization in which all vertices have rational coordinates. Lemma A polytope P is rational ⇔ V + ( I P ) has a rational point. We consider the following point-line arrangement in the plane [Grünbaum, 1967]: x 1 0 x 2 0 x 3 x 4 x 5 x 6 0 x 7 x 8 x 9 0 x 10 0 0 x 11 x 12 x 13 x 14 0 x 15 x 16 x 17 x 18 0 0 x 19 x 20 0 x 21 0 0 x 22 x 23 x 24 S P ( x ) = x 25 0 x 26 x 27 0 x 28 0 0 x 29 0 0 x 30 x 31 x 32 0 x 33 x 34 x 35 0 x 36 0 x 37 x 38 x 39 0 x 40 x 41 0 x 42 x 43 0 x 44 x 45 x 46 0 x 47 0 x 48 x 49 x 50 0 x 51 x 52 x 53 0 Scaling rows and columns to set some variables to 1 (this does not affect rationality): √ 46 + x 46 − 1 ∈ I P ⇒ x 46 = − 1 ± 5 x 2 ⇒ no rational realizations 2 This can be extended to the ideal of the Perles polytope (d=8, v=12, f=34) It is not rational but also its slack ideal is not prime. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 18 / 27
Application 3: Realizability Steinitz problem Check whether an abstract polytopal complex is the boundary of an actual polytope. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 19 / 27
Application 3: Realizability Steinitz problem Check whether an abstract polytopal complex is the boundary of an actual polytope. [Altshuler, Steinberg, 1985]: 4-polytopes and 3-spheres with 8 vertices. The smallest non-polytopal 3-sphere has vertex-facet non-incidence matrix 0 0 0 0 0 x 1 x 2 x 3 x 4 x 5 0 0 0 0 x 6 x 7 0 0 x 8 x 9 0 0 x 10 x 11 x 12 0 0 0 0 x 13 0 0 x 14 x 15 0 0 x 16 x 17 0 0 S P ( x ) = 0 x 18 0 x 19 0 0 0 x 20 x 21 x 22 . x 23 0 x 24 0 0 x 25 x 26 0 0 0 x 27 x 28 0 0 x 29 0 0 0 0 0 x 30 x 31 0 0 0 0 x 32 x 33 x 34 0 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 19 / 27
Application 3: Realizability Steinitz problem Check whether an abstract polytopal complex is the boundary of an actual polytope. [Altshuler, Steinberg, 1985]: 4-polytopes and 3-spheres with 8 vertices. The smallest non-polytopal 3-sphere has vertex-facet non-incidence matrix 0 0 0 0 0 x 1 x 2 x 3 x 4 x 5 0 0 0 0 x 6 x 7 0 0 x 8 x 9 0 0 x 10 x 11 x 12 0 0 0 0 x 13 0 0 x 14 x 15 0 0 x 16 x 17 0 0 S P ( x ) = 0 x 18 0 x 19 0 0 0 x 20 x 21 x 22 . x 23 0 x 24 0 0 x 25 x 26 0 0 0 x 27 x 28 0 0 x 29 0 0 0 0 0 x 30 x 31 0 0 0 0 x 32 x 33 x 34 0 Proposition P is realizable ⇐ ⇒ V + ( I P ) � = ∅ . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 19 / 27
Application 3: Realizability Steinitz problem Check whether an abstract polytopal complex is the boundary of an actual polytope. [Altshuler, Steinberg, 1985]: 4-polytopes and 3-spheres with 8 vertices. The smallest non-polytopal 3-sphere has vertex-facet non-incidence matrix 0 0 0 0 0 x 1 x 2 x 3 x 4 x 5 0 0 0 0 x 6 x 7 0 0 x 8 x 9 0 0 x 10 x 11 x 12 0 0 0 0 x 13 0 0 x 14 x 15 0 0 x 16 x 17 0 0 S P ( x ) = 0 x 18 0 x 19 0 0 0 x 20 x 21 x 22 . x 23 0 x 24 0 0 x 25 x 26 0 0 0 x 27 x 28 0 0 x 29 0 0 0 0 0 x 30 x 31 0 0 0 0 x 32 x 33 x 34 0 Proposition P is realizable ⇐ ⇒ V + ( I P ) � = ∅ . In this case, I P = � 1 � ⇒ no rank 5 matrix with this support ⇒ no polytope. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 19 / 27
Section 4 One more application João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 20 / 27
Dimension of the realization space How much freedom does a certain combinatorial structure give us? João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 21 / 27
Dimension of the realization space How much freedom does a certain combinatorial structure give us? Given a polytope P ⊆ R n , what is the dimension of R ( P )? João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 21 / 27
Dimension of the realization space How much freedom does a certain combinatorial structure give us? Given a polytope P ⊆ R n , what is the dimension of R ( P )? For n = 2, clearly dim ( R ( P )) = 2 v . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 21 / 27
Dimension of the realization space How much freedom does a certain combinatorial structure give us? Given a polytope P ⊆ R n , what is the dimension of R ( P )? For n = 2, clearly dim ( R ( P )) = 2 v . For n = 3 we have dim ( R ( P )) = v + f + 4. [Steinitz] João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 21 / 27
Dimension of the realization space How much freedom does a certain combinatorial structure give us? Given a polytope P ⊆ R n , what is the dimension of R ( P )? For n = 2, clearly dim ( R ( P )) = 2 v . For n = 3 we have dim ( R ( P )) = v + f + 4. [Steinitz] For n > 3 there are very few general results/tools. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 21 / 27
Dimension of the realization space How much freedom does a certain combinatorial structure give us? Given a polytope P ⊆ R n , what is the dimension of R ( P )? For n = 2, clearly dim ( R ( P )) = 2 v . For n = 3 we have dim ( R ( P )) = v + f + 4. [Steinitz] For n > 3 there are very few general results/tools. dim ( R ( P )) ↔ dim ( V + ( I P )) João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 21 / 27
Dimension of the realization space How much freedom does a certain combinatorial structure give us? Given a polytope P ⊆ R n , what is the dimension of R ( P )? For n = 2, clearly dim ( R ( P )) = 2 v . For n = 3 we have dim ( R ( P )) = v + f + 4. [Steinitz] For n > 3 there are very few general results/tools. dim ( R ( P )) ↔ dim ( V + ( I P )) Can we compute the dimension of V ( I P ) ? João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 21 / 27
How to do this? João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 22 / 27
How to do this? Exact Computational Algebra 1 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 22 / 27
How to do this? Exact Computational Algebra 1 Too hard: V ( I P ) has around v × f entries. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 22 / 27
How to do this? Exact Computational Algebra 1 Too hard: V ( I P ) has around v × f entries. Statistical topology from samples 2 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 22 / 27
How to do this? Exact Computational Algebra 1 Too hard: V ( I P ) has around v × f entries. Statistical topology from samples 2 Implies a sufficiently representative sample of polytopes with a given combinatorial structure. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 22 / 27
How to do this? Exact Computational Algebra 1 Too hard: V ( I P ) has around v × f entries. Statistical topology from samples 2 Implies a sufficiently representative sample of polytopes with a given combinatorial structure. Hopeless in general. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 22 / 27
How to do this? Exact Computational Algebra 1 Too hard: V ( I P ) has around v × f entries. Statistical topology from samples 2 Implies a sufficiently representative sample of polytopes with a given combinatorial structure. Hopeless in general. However Maybe we can use the structure of the variety to do enough? 3 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 22 / 27
Perturbing a polytope Let us go to a related more basic problem: How to perturb a polytope while preserving the combinatorics? João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 23 / 27
Perturbing a polytope Let us go to a related more basic problem: How to perturb a polytope while preserving the combinatorics? Given a polytope P , we can always add noise to the entries of S P but then we are away from V ( I P ) . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 23 / 27
Perturbing a polytope Let us go to a related more basic problem: How to perturb a polytope while preserving the combinatorics? Given a polytope P , we can always add noise to the entries of S P but then we are away from V ( I P ) . Can we project it back? João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 23 / 27
Perturbing a polytope Let us go to a related more basic problem: How to perturb a polytope while preserving the combinatorics? Given a polytope P , we can always add noise to the entries of S P but then we are away from V ( I P ) . Can we project it back? Yes!!! By using the fact that V ( I P ) = { X : rank ( X ) ≤ d + 1 } ∩ L . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 23 / 27
Perturbing a polytope Let us go to a related more basic problem: How to perturb a polytope while preserving the combinatorics? Given a polytope P , we can always add noise to the entries of S P but then we are away from V ( I P ) . Can we project it back? Yes!!! By using the fact that V ( I P ) = { X : rank ( X ) ≤ d + 1 } ∩ L . Proto-theorem - GPP sometime in the future In general, Dykstra’s alternate projection algorithm will applied to ¯ S = S P + noise will converge to the projection of ¯ S in V ( I P ) . João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 23 / 27
Perturbing a polytope Let us go to a related more basic problem: How to perturb a polytope while preserving the combinatorics? Given a polytope P , we can always add noise to the entries of S P but then we are away from V ( I P ) . Can we project it back? Yes!!! By using the fact that V ( I P ) = { X : rank ( X ) ≤ d + 1 } ∩ L . Proto-theorem - GPP sometime in the future In general, Dykstra’s alternate projection algorithm will applied to ¯ S = S P + noise will converge to the projection of ¯ S in V ( I P ) . This is not a full answer to the question, but might be enough. João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 23 / 27
Enter the statistics Idea: João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 24 / 27
Enter the statistics Idea: Start with S P ∈ V R ( I P ) ; 1 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 24 / 27
Enter the statistics Idea: Start with S P ∈ V R ( I P ) ; 1 Add noise to each entry following N ( 0 , ǫ ) distribution; 2 João Gouveia (UC ) Slack variety of a polytope and its applications ICERM 2018 24 / 27
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