Algebraic and combinatorial methods for bounding the number of the complex embeddings of minimally rigid graphs E. Bartzos, I.Z. Emiris, J. Schicho 14 June 2019 Geometric constraint systems: rigidity, flexibility and applications Lancaster University A R C A D E S
Counting realizations – Existing work • Number of realizations and asymptotic bounds ▶ Complex embeddings for Laman graphs (Capko, Gallet, Grasseger, Koutschan, Lubbes, Schicho) ▶ Complex embeddings for Geiringer graphs and asymptotic lower bounds (Grasegger, Koutschan, Tsigaridas) ▶ Upper bounds (Borcea, Streinu) ▶ Mixed volume methods for Geiringer and Laman graphs(Emiris, Tsigaridas, Varvitsiotis) ▶ Real embeddings for Geiringer and Laman graphs and real bounds for specific graphs (EB, Emiris, Legersky, Tsigaridas) 1
Bounds on the number of complex solutions • Fast computation methods • Homotopy continuation solvers • Tight upper bounds on the number of realizations • Asymptotic upper bounds 2
Bézout bound – Projective & multi-projective case Bézout bound : ∏ m i = 1 deg ( f i ) 3
Bézout bound – Projective & multi-projective case Bézout bound : ∏ m i = 1 deg ( f i ) Multihomogeneous Bézout bound : Let X 1 , X 2 , . . . , X k be a partition of the m variables, m i = | X i | and d ij be the degree of the i-th equation in th j-th set of variables. Then the number of solutions of the system is bounded from above by the coefficient of · · · X m k the monomial X m 1 · X m 2 in the polynomial 1 2 k m ∏ ( d i 1 · X 1 + d i 2 · X 2 + . . . d ik · X k ) i = 1 3
Bézout bound – Projective & multi-projective case Bézout bound : ∏ m i = 1 deg ( f i ) Multihomogeneous Bézout bound : Let X 1 , X 2 , . . . , X k be a partition of the m variables, m i = | X i | and d ij be the degree of the i-th equation in th j-th set of variables. Then the number of solutions of the system is bounded from above by the coefficient of · · · X m k the monomial X m 1 · X m 2 in the polynomial 1 2 k m ∏ ( d i 1 · X 1 + d i 2 · X 2 + . . . d ik · X k ) i = 1 Example: f = xy − 1, g = x 2 − 1 Bézout = 4 coeff ( XY , ( X + Y ) · ( 2 X )) = 2 3
Newton polytopes and mixed volumes Definition (Newton polytope) Let f be a polynomial in C [ x 1 , . . . , x n ] such that f = Σ c a x a , where x a = x a 1 1 · x a 2 2 . . . x a n n . The Newton polytope of f is the convex hull of the set of the exponents of the monomials with non-zero coefficients. 4
Newton polytopes and mixed volumes Definition (Newton polytope) Let f be a polynomial in C [ x 1 , . . . , x n ] such that f = Σ c a x a , where x a = x a 1 1 · x a 2 2 . . . x a n n . The Newton polytope of f is the convex hull of the set of the exponents of the monomials with non-zero coefficients. y Example: 3 f = x 3 y + y 3 − 2 xy + 5 x − 3 2 S = { ( 3 , 1 ) , ( 0 , 3 ) , ( 1 , 1 ) , ( 1 , 0 ) , ( 0 , 0 ) } 1 NP ( F ) = ConvHull ( S ) x 0 1 2 3 4
Newton polytopes and mixed volumes Minkowski addition P 1 + P 2 = { p 1 + p 2 | p 1 ∈ P 1 , p 2 ∈ P 2 } y 3 2 P 1 y 1 5 x 0 4 1 2 3 3 y P 1 + P 2 2 3 1 2 x 0 P 2 1 1 2 3 4 5 6 x 0 1 2 3 5
Newton polytopes and mixed volumes Mixed volume MV ( P 1 , P 2 , . . . , P n ) is the coefficient of λ 1 · λ 2 . . . λ n in the homogeneous polynomial Vol n ( λ 1 P 1 + λ 2 P 2 + · · · + λ n P n ) 6
Newton polytopes and mixed volumes Mixed volume MV ( P 1 , P 2 , . . . , P n ) is the coefficient of λ 1 · λ 2 . . . λ n in the homogeneous polynomial Vol n ( λ 1 P 1 + λ 2 P 2 + · · · + λ n P n ) Theorem (Bernstein, Khovanskii, Kushnirenko) The number of roots of a system of polynomials f 1 , f 2 , . . . , f n in ( C ∗ ) n is bounded from above by the mixed volume of the Newton polytopes of these polynomials. 6
Relations between complex bounds # complex solutions ≤ Mixed Volume ≤ m-Bézout ≤ Bézout 7
Relations between complex bounds # complex solutions ≤ Mixed Volume ≤ m-Bézout ≤ Bézout • Mixed volume is tight for generic coefficients. • MV = Bézout for simplices. • MV = m-Bézout subsets of boxes { 0 , d 1 } × { 0 , d 2 } × · · · × { 0 , d n } , that verify maximum degree for all the sets of the monomials. y y f 2 = x + y 2 x + y 2 + 5 2 f 1 = x 2 + y 2 − 3 2 1 1 x x 0 0 1 2 1 2 7
Algebraic Modelling – Sphere equations Fix coordinates of an edge (2d and S 2 case) or a triangle (3d case) to remove rigid motions. ∥ X u − X v ∥ 2 = λ 2 uv ∀ uv ∈ E , • c ∗ ( G ) = # of complex embeddings Introduce sphere equations ∥ X u ∥ 2 = s u ∀ u ∈ V , s u + s v − 2 ⟨ X u , X v ⟩ = λ 2 uv ∀ uv ∈ E , • s u = const. in the case of S n • Structure of equations leads to sharper complex bounds. 8
m-Bézout bound for sphere equations ∥ X u ∥ 2 = s u ∀ u ∈ V , s u + s v − 2 ⟨ X u , X v ⟩ = λ 2 uv ∀ uv ∈ E , (natural) partition of variables: X i = { x i 1 , . . . , x i d , s i } m-Bézout for minimally rigid graphs: | E ′ | n − d ∏ ∏ 2 X i · ( X k 1 + X k 2 ) i = 1 k = 1 where E ′ = E − { simplex } . We need to compute the coefficient of X d + 1 · X d + 1 · · · X d + 1 . 1 2 n 9
m-Bézout bound for sphere equations ∥ X u ∥ 2 = s u ∀ u ∈ V , s u + s v − 2 ⟨ X u , X v ⟩ = λ 2 uv ∀ uv ∈ E , (natural) partition of variables: X i = { x i 1 , . . . , x i d , s i } product for edge equations: | E ′ | (˜ X k 1 + ˜ ∏ X k 2 ) k = 1 We need to compute the coefficient of ˜ 1 · ˜ 2 · · · ˜ X d X d X d n . The m-Bézout bound is 2 n − d · coeff. 9
Combinatorial Algorithm for m-Bézout bound Theorem Let H ( V , E ′ ) be a graph obtained after removing the fixed ( d − 1 ) − simplex from G ( V , E ) . We define H as the set of all directed graphs such that if we remove the orientation, they coincide with H and each non-fixed vertex has indegree d. Then, the coefficient of the monomial ˜ 1 · ˜ 2 · · · ˜ X d X d X d m of the previous product is exactly the same as |H| . 10
Combinatorial Algorithm for m-Bézout bound 2 orientations ∗ 2 6 − 2 = 32 11
Combinatorial Algorithm for m-Bézout bound 2 orientations ∗ 2 6 − 2 = 32 c 2 ( G ) = 24 , c S 2 ( G ) = 32 11
Permanent of a matrix as m-Bézout bound ( 1 , 3 ) ( 2 , 3 ) ( 1 , 5 ) ( 2 , 6 ) ( 3 , 4 ) ( 4 , 5 ) ( 4 , 6 ) ( 5 , 6 ) 1 1 0 0 1 0 0 0 x 3 1 1 0 0 1 0 0 0 y 3 0 0 0 0 1 1 1 0 x 4 0 0 0 0 1 1 1 0 y 4 0 0 1 0 0 1 0 1 x 5 0 0 1 0 0 1 0 1 y 5 0 0 0 1 0 0 1 1 x 6 0 0 0 1 0 0 1 1 y 6 12
Permanent of a matrix as m-Bézout bound m ( − 1 ) m −| M | ∑ ∏ ∑ per ( A ) = (1) a ij i = 1 j ∈ M M ⊆{ 1 , 2 ,..., m } What is interesting for us is that there is a relation between the per ( A ) and the m-Bézout bound: Theorem 1 m-Bézout = m 1 ! m 2 ! . . . m k ! · per ( A ) In the case of sphere equations we get the following: ( 2 ) n − d · per ( A ) d ! • Current asymptotic bounds for permanent do not ameliorate Bézout bounds. 13
Runtimes n Laman graphs comb. MV Maple ’s m-Bézout c 2 ( G ) permanent phcpy Python 6 0.0096s 0.0242s 0.114s 0.0123s 7 0.01526s 0.104s 0.12s 0.0152s 8 0.0276s 0.163s 0.138s 0.0431s 9 0.066s 0.397s 0.26s 0.076s 10 0.1764s 1.17s 0.302s 0.148s 11 0.5576s 2.84s 0.4s 0.2761s 12 6.35995s 11.7s 0.897s 0.5623s 18 17h 5min 27s 3h* 454.5s 6.84s 14
Runtimes n Geiringer graphs MV Maple ’s m-Bézout phcpy solver permanent phcpy Python 6 0.652s 0.107s 0.113 0.0098s 7 3.01s 0.175s 0.256 0.02s 8 20.1s 3.48s 0.359 0.0492s 9 2min 33s 2 min 16s 0.406s 0.149s 10 16min 1s 1h 58min 16s 1.127s 0.338s 11 2h 13min 51s > 1.5 day 3.033s 0.442s 12 - >6 days* 32.079s 1.3s • Can we find an optimal simplex? 14
Relation between m-Bézout bound and mixed volume and num- ber of complex embeddings • Mixed volume= m-Bézout bound in almost every example. • Mixed volume= m-Bézout in every reduced edge equation system ⟨ X u , X v ⟩ = const . ∀ uv ∈ E . • Missing terms from the m-Bézout Newton polytope for Laman graphs: x u + y u + x v + y v + x u · y v + y u · x v + x u · x v + y u · y v = const . 15
Comparing m-Bézout bound and the c ( G ) • Spatial embeddings=m-Bézout bound for every planar minimally rigid graph in C 3 . • Spherical embeddings for planar graphs. n mBézout c 2 ( G ) c S 2 ( G ) 6 32 24 32 7 64 56 64 8 192 136 192 9 512 344 512 10 1536 880 1536 11 4096 2288 4096 12 15630 6180 8704 16
Determinantal conditions for mixed volume Definition (Polytope intersections & Initial forms) Let S and w be respectively a polytope and a non-zero vector in R n . We denote the subset of S that minimizes the inner product ⟨· , w ⟩ as S w . c q x q in the direction of w The initial form of a polynomial f = ∑ q ∈ S is f w = q ∈ S w c q x q . ∑ Theorem (Bernstein’s second theorem) The number of roots of a system of polynomials f 1 , f 2 , . . . , f n in ( C ∗ ) n is exactly mixed volume of their Newton polytopes iff ∀ α ∈ R n the system f w 1 , f w 2 , . . . , f w has no solutions in ( C ∗ ) n . n 17
Exactness of m-Bézout bound • The determinantal conditions can be verified by checking all the w ∈ R n that are inner normals of the faces of P 1 + P 2 + · · · + P n . • Normals of the facets and a subset of their linear combinations. 18
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