Algebraic and combinatorial methods for bounding the number of the - - PowerPoint PPT Presentation

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(fi) 3

Bézout bound – Projective & multi-projective case

Bézout bound: ∏m

i=1 deg(fi)

Multihomogeneous Bézout bound: Let X1, X2, . . . , Xk be a partition of the m variables, mi = |Xi| and dij 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 the monomial X m1

1

· X m2

2

· · · X mk

k

in the polynomial

m

∏

i=1

(di1 · X1 + di2 · X2 + . . . dik · Xk)

3

Bézout bound – Projective & multi-projective case

Bézout bound: ∏m

i=1 deg(fi)

Multihomogeneous Bézout bound: Let X1, X2, . . . , Xk be a partition of the m variables, mi = |Xi| and dij 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 the monomial X m1

1

· X m2

2

· · · X mk

k

in the polynomial

m

∏

i=1

(di1 · X1 + di2 · X2 + . . . dik · Xk) Example: f = xy − 1, g = x2 − 1 Bézout = 4 coeff (XY , (X + Y ) · (2X)) = 2

3

Newton polytopes and mixed volumes

Definition (Newton polytope) Let f be a polynomial in C[x1, . . . , xn] such that f = Σcaxa, where xa = xa1

1 · xa2 2 . . . xan 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[x1, . . . , xn] such that f = Σcaxa, where xa = xa1

1 · xa2 2 . . . xan n .

The Newton polytope of f is the convex hull of the set of the exponents of the monomials with non-zero coefficients. Example: f = x3y + y3 − 2xy + 5x − 3 S = {(3, 1), (0, 3), (1, 1), (1, 0), (0, 0)} NP(F) = ConvHull(S) y x 1 2 3 1 2 3

4

Newton polytopes and mixed volumes

Minkowski addition P1 + P2 = {p1 + p2|p1 ∈ P1, p2 ∈ P2} y x 1 2 3 1 2 3 P1 y x 1 2 3 1 2 3 P2 y x 1 2 3 4 5 1 2 3 4 5 6 P1 + P2

5

Newton polytopes and mixed volumes

Mixed volume MV (P1, P2, . . . , Pn) is the coefficient of λ1 · λ2 . . . λn in the homogeneous polynomial Voln(λ1P1 + λ2P2 + · · · + λnPn)

6

Newton polytopes and mixed volumes

Mixed volume MV (P1, P2, . . . , Pn) is the coefficient of λ1 · λ2 . . . λn in the homogeneous polynomial Voln(λ1P1 + λ2P2 + · · · + λnPn) Theorem (Bernstein, Khovanskii, Kushnirenko) The number of roots of a system of polynomials f1, f2, . . . , fn 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, d1} × {0, d2} × · · · × {0, dn}, that verify maximum degree for all the sets of the monomials. y x 1 2 1 2 f1 = x2 + y2 − 3 y x 1 2 1 2 f2 = x + y2x + y2 + 5

7

Algebraic Modelling – Sphere equations

Fix coordinates of an edge (2d and S2 case) or a triangle (3d case) to remove rigid motions. ∥Xu − Xv∥2 = λ2

uv ∀uv ∈ E ,

• c∗(G) = # of complex embeddings

Introduce sphere equations ∥Xu∥2 = su ∀u ∈ V , su + sv − 2⟨Xu, Xv⟩ = λ2

uv ∀uv ∈ E ,

• su = const. in the case of Sn
• Structure of equations leads to sharper complex bounds.

8

m-Bézout bound for sphere equations

∥Xu∥2 = su ∀u ∈ V , su + sv − 2⟨Xu, Xv⟩ = λ2

uv ∀uv ∈ E ,

(natural) partition of variables: Xi = {xi1, . . . , xid, si} m-Bézout for minimally rigid graphs:

n−d

∏

i=1

2Xi ·

|E ′|

∏

k=1

(Xk1 + Xk2) where E ′ = E − {simplex}. We need to compute the coefficient of X d+1

1

· X d+1

2

· · · X d+1

n

.

9

m-Bézout bound for sphere equations

∥Xu∥2 = su ∀u ∈ V , su + sv − 2⟨Xu, Xv⟩ = λ2

uv ∀uv ∈ E ,

(natural) partition of variables: Xi = {xi1, . . . , xid, si} product for edge equations:

|E ′|

∏

k=1

(˜ Xk1 + ˜ Xk2) We need to compute the coefficient of ˜ X d

1 · ˜

X d

2 · · · ˜

X d

n .

The m-Bézout bound is 2n−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 ˜ X d

1 · ˜

X d

2 · · · ˜

X d

m of the previous

product is exactly the same as |H|.

10

Combinatorial Algorithm for m-Bézout bound

2 orientations ∗ 26−2 = 32

11

Combinatorial Algorithm for m-Bézout bound

2 orientations ∗ 26−2 = 32 c2(G) = 24 , cS2(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) x3 1 1 1 y3 1 1 1 x4 1 1 1 y4 1 1 1 x5 1 1 1 y5 1 1 1 x6 1 1 1 y6 1 1 1

12

Permanent of a matrix as m-Bézout bound

per (A) =

∑

M⊆{1,2,...,m}

(−1)m−|M|

m

∏

i=1

∑

j∈M

aij (1) What is interesting for us is that there is a relation between the per(A) and the m-Bézout bound: Theorem m-Bézout = 1 m1!m2! . . . mk! · per (A) In the case of sphere equations we get the following:

( 2

d!

)n−d

· per (A)

• Current asymptotic bounds for permanent do not ameliorate

Bézout bounds.

13

Runtimes

n Laman graphs comb. MV Maple’s m-Bézout c2(G) phcpy permanent 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 phcpy MV Maple’s m-Bézout solver phcpy permanent 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 ⟨Xu, Xv⟩ = const. ∀uv ∈ E .

• Missing terms from the m-Bézout Newton polytope

for Laman graphs: xu + yu + xv + yv + xu · yv + yu · xv+xu · xv + yu · yv = const.

15

Comparing m-Bézout bound and the c(G)

• Spatial embeddings=m-Bézout bound for every planar

minimally rigid graph in C3.

• Spherical embeddings for planar graphs.

n mBézout c2(G) cS2(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

• Rn. We denote the subset of S that minimizes the inner product

⟨·, w⟩ as Sw. The initial form of a polynomial f = ∑

q∈S

cqxq in the direction of w is f w =

∑

q∈Sw cqxq.

Theorem (Bernstein’s second theorem) The number of roots of a system of polynomials f1, f2, . . . , fn in (C∗)n is exactly mixed volume of their Newton polytopes iff ∀α ∈ Rn the system f w

1 , f w 2 , . . . , f w n

has no solutions in (C∗)n.

17

Exactness of m-Bézout bound

• The determinantal conditions can be verified by checking all

the w ∈ Rn that are inner normals of the faces of P1 + P2 + · · · + Pn.

• Normals of the facets and a subset of their linear

combinations.

18

Exactness of m-Bézout bound

• The determinantal conditions can be verified by checking all

the w ∈ Rn that are inner normals of the faces of P1 + P2 + · · · + Pn.

• Normals of the facets and a subset of their linear

combinations.

• m-Bézout bound is sharp iff Bernstein’s second theorem can

be applied for m-Bézout polytopes.

• m-Bézout polytopes are simpler than Newton polytopes of

sphere equations.

18

Exactness of m-Bézout bound

• Desargues graph as test case

(mBe = 32, c2(G) = 24, cS2(G) = 32)

• Find the number of complex embeddings for cases in 3d in

which no other method can be applied.

• Find if the conjecture about planar Geiringer graphs is right.

19

(More) open questions

• Indegree-constrained orientations of graphs
• Sparse permanents – Perfect matchings of bipartite graphs
• Algebraic formulation for graphs on other surfaces.

20

Thank you

vbartzos@di.uoa.gr emiris@di.uoa.gr users.uoa.gr/~vbartzos cgi.di.uoa.gr/~emiris/ josef.schicho@risc.jku.at www3.risc.jku.at/people/jschicho/

21