Real monodromy action Jonathan Hauenstein Margaret H. Regan ICERM - PowerPoint PPT Presentation

Real monodromy action Jonathan Hauenstein Margaret H. Regan ICERM Workshop on Monodromy and Galois Groups in Enumerative Geometry and Applications - Sept. 2, 2020

Outline Background 1 Motivation Complex monodromy group Real monodromy group 2 Real monodromy structure 3 3RPR mechanism Summary 4

Motivation The complex monodromy group encodes information regarding the permutations of solutions to a polynomial system over loops in the parameter space. It gives structural information in the following ways: symmetry of solutions some restrictions to number of real solutions decomposition of varieties into irreducible components

Motivation Main question: How can we understand the behavior of real solutions over real loops in parameter space? This idea influences many applications: in kinematics, it is related to nonsingular assembly mode change for parallel manipulators.

Complex monodromy group Fix a generic basepoint Assign an ordering of the solutions Pick a loop in the parameter space that avoids singularities How do the solutions permute along the loop? The collection of the permutations is the complex monodromy group . Note: The complex monodromy group is independent of choice of basepoint and has an equivalent monodromy group when a general curve section of the parameter space is considered.

Homotopy continuation How do we take these loops? H p z, t q “ F p z ; t ¨ p 0 ` p 1 ´ t q ¨ p q F p z ; p 0 q : start system F p z ; p q : target system

Example Consider the parameterized polynomial system „ x 2 1 ´ x 2  2 ´ p 1 F p x ; p q “ “ 0 . 2 x 1 x 2 ´ p 2 Take basepoint b “ p 1 , 0 q P C 2 such that p 2 1 ` p 2 2 ‰ 0 Order the 4 nonsingular isolated solutions: x p 1 q “ p 1 , 0 q , x p 2 q “ p´ 1 , 0 q , x p 3 q “ p 0 , ?´ 1 q , x p 4 q “ p 0 , ´?´ 1 q Restrict parameter space to the line parametrized by ℓ p t q “ p 1 ´ t, 2 t q This gives 2 singular points, t ˘ Loop around these singular points gives us two permutations: σ γ ` “ p 1 3 qp 2 4 q and σ γ ´ “ p 1 4 qp 2 3 q These generate the Klein group on four elements K 4 “ Z 2 ˆ Z 2 Ă S 4

Example

Real monodromy group Fix a real basepoint Assign an ordering of the real solutions Pick a real loop in the real parameter space that avoids singularities How do the solutions permute along the loop? The collection of the permutations is the real monodromy group . Note: This definition has restrictions: (1) only basepoint independent within the same connected component and (2) it’s not clear how to slice.

Example 1 Consider the parameterized polynomial system „ x 2 1 ´ x 2  2 ´ p 1 F p x ; p q “ “ 0 . 2 x 1 x 2 ´ p 2 Take basepoint b “ p 1 , 0 q P R 2 such that p 2 1 ` p 2 2 ‰ 0 Order the 2 real nonsingular isolated solutions: x p 1 q “ p 1 , 0 q , x p 2 q “ p´ 1 , 0 q Loop around the singular point gives us the permutation: σ γ “ p 1 2 q Thus, the real monodromy group is S 2 “ tp 1 q , p 1 2 qu .

Example 2 Consider a slightly modified parameterized polynomial system „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 . 2 x 1 x 2 ´ p 2 Take basepoint b “ p´ 1 , 0 q P R 2 such that p 2 1 ` p 2 2 ‰ 0 Order the real 4 nonsingular isolated solutions: x p 1 q “ p 1 , 0 q , x p 2 q “ p´ 1 , 0 q , x p 3 q “ p 0 , 1 q , x p 4 q “ p 0 , ´ 1 q No nontrivial real loop exists around the singularity for all 4 solutions Fundamental group is trivial Thus, the real monodromy group is trivial

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: Consider x p 1 q “ p 1 , 0 q along the loop shown. Does it stay real and nonsingular along the loop?

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: Consider x p 1 q “ p 1 , 0 q along the loop shown. Does it stay real and nonsingular along the loop? Yes

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: Consider x p 1 q “ p 1 , 0 q along the loop shown. Does it stay real and nonsingular along the loop? Yes Does the solution permute?

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: Consider x p 1 q “ p 1 , 0 q along the loop shown. Does it stay real and nonsingular along the loop? Yes Does the solution permute? Yes, to x p 2 q “ p´ 1 , 0 q

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: Consider x p 1 q “ p 1 , 0 q along the loop shown. Does it stay real and nonsingular along the loop? Yes Does the solution permute? Yes, to x p 2 q “ p´ 1 , 0 q We represent this as: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others |

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others | In general,we have: G k : k -ordered solutions Ñ sets of k -ordered solutions that can be attained by a real loop where all solutions in the set remain real and nonsingular.

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others | Next, consider the set of sols. t x p 1 q , x p 2 q u . Do these both stay real and nonsingular along the loop?

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others | Next, consider the set of sols. t x p 1 q , x p 2 q u . Do these both stay real and nonsingular along the loop? Yes

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others | Next, consider the set of sols. t x p 1 q , x p 2 q u . Do these both stay real and nonsingular along the loop? Yes Do any permutations occur?

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others | Next, consider the set of sols. t x p 1 q , x p 2 q u . Do these both stay real and nonsingular along the loop? Yes Do any permutations occur? Yes t x p 1 q , x p 2 q u Ñ t x p 2 q , x p 1 q u

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Let’s compute the real monodromy structure: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others | Continuing with all pairs, we obtain: G 2 t 1 , 2 u Ñ tt 1 , 2 u , t 2 , 1 uu | t q 1 , q 2 u Ñ tt q 1 , q 2 uu for all others |

Real monodromy structure „ p x 2 1 ´ x 2 2 ´ p 1 qp x 2  1 ` p 1 q F p x ; p q “ “ 0 2 x 1 x 2 ´ p 2 Continuing in this fashion, the real monodromy structure is: G 1 t 1 u , t 2 u Ñ tt 1 u , t 2 uu | t q 1 u Ñ tt q 1 uu for all others | G 2 t 1 , 2 u Ñ tt 1 , 2 u , t 2 , 1 uu | t q 1 , q 2 u Ñ tt q 1 , q 2 uu for all others | G 3 t q 1 , q 2 , q 3 u Ñ tt q 1 , q 2 , q 3 uu | G 4 t q 1 , q 2 , q 3 , q 4 u Ñ tt q 1 , q 2 , q 3 , q 4 uu |

3RPR mechanism φ 2 1 ` φ 2 2 ´ 1 » fi p 2 1 ` p 2 2 ´ 2 p a 3 p 1 ` b 3 p 2 q φ 1 ` 2 p b 3 p 1 ´ a 3 p 2 q φ 2 — ffi — ` a 2 3 ` b 2 ffi 3 ´ c 1 — ffi 2 ´ 2 A 2 p 1 ` p a 2 ´ a 3 q 2 ` b 2 — ffi p 2 1 ` p 2 3 ` A 2 F p p, φ ; c q “ 2 ´ c 2 — ffi — ffi ` 2 pp a 2 ´ a 3 q p 1 ´ b 3 p 2 ` A 2 a 3 ´ A 2 a 2 q φ 1 — ffi — ffi ` 2 p b 3 p 1 ` p a 2 ´ a 3 q p 2 ´ A 2 b 3 q φ 2 – fl p 2 1 ` p 2 2 ´ 2 p A 3 p 1 ` B 3 p 2 q ` A 2 3 ` B 2 3 ´ c 3 Fix c 3 “ 100 and consider ℓ 1 and ℓ 2 as paramters. At the “home” position c ˚ “ p 75 , 70 q , the system F p p, φ ; c ˚ q “ 0 has 6 nonsingular real solutions.

3RPR mechanism x p 1 q x p 2 q x p 3 q x p 4 q x p 5 q x p 6 q The 6 solutions to F p p, φ ; c ˚ q “ 0 .

3RPR mechanism (a) (b) Regions of the parameter space c “ p c 1 , c 2 q colored by the number of real solutions where (a) is the full view and (b) is a zoomed in view of the lower left corner. The navy blue region has 6 real solutions, the grey blue region has 4 real solutions, the baby blue region has 2 real solutions, and the white region has 0 real solutions.

3RPR mechanism Illustration of a nonsingular assembly mode change between x p 4 q and x p 5 q .