Computer algebra algorithms for solving polynomial systems, sofware - PowerPoint PPT Presentation

Computer algebra algorithms for solving polynomial systems, sofware and applications Thibaut Verron Johannes Kepler University, Institute for Algebra, Linz, Austria 1

Non-linear modelization and computer algebra Applications Robotics Cryptography Dynamical systems ... Many tools: System of polynomial ◮ Numeric equations (and inequations) ◮ Ex: Newton iteration, homotopy... ◮ Trade precision for speed ◮ Computer Algebra ◮ Ex: Resultants, Gröbner bases... ◮ Exact, exhaustive and certifiable ◮ Hybrid Solutions 2

Non-linear modelization and computer algebra Applications Robotics Cryptography Dynamical systems ... Many tools: System of polynomial ◮ Numeric equations (and inequations) ◮ Ex: Newton iteration, homotopy... ◮ Trade precision for speed ◮ Computer Algebra ◮ Ex: Resultants, Gröbner bases... ◮ Exact, exhaustive and certifiable ◮ Hybrid Solutions 2

Non-linear modelization and computer algebra Applications Robotics Cryptography Dynamical systems ... Many tools: System of polynomial ◮ Numeric equations (and inequations) ◮ Ex: Newton iteration, homotopy... ◮ Trade precision for speed ◮ Computer Algebra ◮ Ex: Resultants, Gröbner bases... ◮ Exact, exhaustive and certifiable ◮ Hybrid Solutions 2

Generic and structured systems Goal: exact, exhaustive and certified results ◮ Replace or supplement numeric calculations with symbolic manipulations ◮ Difficulty: intrinsic complexity of the objects being computed Examples: ◮ NP-complete problem over finite fields ◮ Bézout bound: number of solutions exponential (product of the degrees) ◮ Worst case: doubly exponential space complexity [Mayr, Meyer 1984] ◮ For generic system, singly exponential bounds (time and space) In practice, systems from applications are... ◮ ... not generic ◮ ... not instances of the worst case complexity Key question : identify underlying structures to recover the generic complexity 3

An example: algebraic classification for magnetic resonance imagery (With B. Bonnard, J.-C. Faugère, A. Jacquemard and M. Safey El Din) ◮ Context: Magnetic Resonance Imagery ◮ Goal: optimize contrast Optimized Bad contrast ◮ Optimal control approach: the Bloch model y i , z i : 2 n dynamic variables Bloch ball: y 2 i + z 2 i ≤ 1 d  = − Γ i y i − u z i d t y i    ( i = 1 , 2 , . . . , n ) d  = − γ i ( 1 − z i ) + u y i d t z i   u : control function γ i , Γ i : 2 n physical parameters fixed by the experimental seting γ i > 0, Γ i > 0, 2 Γ i ≥ γ i 4

Semi-algebraic classification problem for MRI z • N σ N + σ − Problem: classification of optimal trajectories σ N + σ h s σ − ➤ Σ 1 ➤ ◮ Control of a single particle: done σ N + σ h s σ + σ − ➤ − 1 σ N + σ h s σ b + σ v s σ − ◮ For two particles: more complicated O y ➤ • ➤ Σ 4 ◮ Classify some algebraic invariants instead Σ 3 • S ′ S 1 S 2 2 • • • ➤ Σ 2 S 3 σ N + σ h ◮ Used for choosing simulations to run s σ + ➤ bridge: σ b + +1 σ N + σ h s σ + : ∅ if S 1 = S 2 Example of algebraic invariant: ◮ Linked to equilibrium points ◮ Equations: X � D = ∂ D ∂ y 1 = ∂ D ∂ z 1 = ∂ D ∂ y 2 = ∂ D � B V = ∂ z 2 = 0 1 V 1 ◮ D : determinant of 4 vector fields y 2 i + z 2 ◮ Inequalities: B = 2 � i ≤ 1 � 2 ◮ Classification question: real points of V ∩ B 3 3 4 2 2 2 depending on γ i , Γ i Γ 5

Results for the MRI classification problem State of the art: ◮ Existing tools can’t solve the problem efficiently ◮ 1000s on the case of water (easier: γ 1 =Γ 1 = 1), full problem out of reach ◮ Complicated output for further steps Results: ◮ Dedicated algorithm exploiting the structure of the system (determinants of matrices) ◮ Implemented in Maple ◮ Used to give full classification to the application ◮ 10s on the case of water, 4h on the full problem Tools: ◮ Real geometry: Whitney stratification, Thom’s isotopy theorem, critical points ◮ Algebra: determinantal ideals, incidence varieties ◮ Computer Algebra: polynomial elimination 6

Classification in the case of water ( γ 1 = Γ 1 = 1) Γ 2 Γ 2 30 4 Fat 1 1 1 1 1 1 3 1 3 20 Cerebrospinal fluid 3 1 1 2 2 10 1 2 1 2 1 3 1 0 0 γ 2 γ 2 0 2 4 6 8 10 12 14 16 0 1 2 3 4 5 6 Γ 2 Γ 2 1 . 6 0 . 7 3 1 . 4 1 0 . 68 1 1 1 . 2 2 0 . 66 1 2 0 . 64 1 0 . 8 1 1 0 . 6 0 . 62 3 3 1 0 . 4 0 . 6 γ 2 γ 2 0 . 78 0 . 79 0 . 8 0 . 81 0 . 82 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 7

Classification in the case of water ( γ 1 = Γ 1 = 1) Γ 2 Γ 2 30 4 Fat 1 1 1 1 1 1 3 1 3 20 Cerebrospinal fluid 3 1 1 2 2 10 1 2 1 2 1 3 1 0 0 γ 2 γ 2 0 2 4 6 8 10 12 14 16 0 1 2 3 4 5 6 Γ 2 Γ 2 1 . 6 0 . 7 3 1 . 4 1 0 . 68 1 1 1 . 2 2 0 . 66 1 2 0 . 64 1 0 . 8 1 1 0 . 6 0 . 62 3 3 1 0 . 4 0 . 6 γ 2 γ 2 0 . 78 0 . 79 0 . 8 0 . 81 0 . 82 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 7

Classification in the case of water ( γ 1 = Γ 1 = 1) Γ 2 Γ 2 30 4 Fat 1 1 1 1 1 1 3 1 3 20 Cerebrospinal fluid 3 1 1 2 2 10 1 2 1 2 1 3 1 0 0 γ 2 γ 2 0 2 4 6 8 10 12 14 16 0 1 2 3 4 5 6 Γ 2 Γ 2 1 . 6 0 . 7 3 1 . 4 1 0 . 68 1 1 1 . 2 2 0 . 66 1 2 0 . 64 1 0 . 8 1 1 0 . 6 0 . 62 3 3 1 0 . 4 0 . 6 γ 2 γ 2 0 . 78 0 . 79 0 . 8 0 . 81 0 . 82 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 7

Classification in the case of water ( γ 1 = Γ 1 = 1) Γ 2 Γ 2 30 4 Fat 1 1 1 1 1 1 3 1 3 20 Cerebrospinal fluid 3 1 1 2 2 10 1 2 1 2 1 3 1 0 0 γ 2 γ 2 0 2 4 6 8 10 12 14 16 0 1 2 3 4 5 6 Γ 2 Γ 2 1 . 6 0 . 7 3 1 . 4 1 0 . 68 1 1 1 . 2 2 0 . 66 1 2 0 . 64 1 0 . 8 1 1 0 . 6 0 . 62 3 3 1 0 . 4 0 . 6 γ 2 γ 2 0 . 78 0 . 79 0 . 8 0 . 81 0 . 82 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 7

Main computer algebra building block : polynomial elimination Polynomial elimination: ◮ Given an ideal I ⊂ K [ X 1 , . . . , X n , G 1 , . . . , G r ] ◮ Compute a basis of I G = I ∩ K [ G 1 , . . . , G r ] Computing eliminations allows to... ◮ ... compute projections of varieties ◮ ... solve if finitely many solutions (by iterating) ◮ ... compute unions and differences of varieties (by lifing) Many tools: resultants, triangular sets, Gröbner bases  2 X 1 X 2 + 4 G 1 G 2 − 8 G 2      2 G 1 − 3 X 2 2 G 2 − 3 G 2 2  2 X 1 X 1 G 1 + 2 X 2 G 2        3 + 3 X 1 G 2 2 − 12 X 2 G 1 G 2 2 + 56 X 2 G 2 3 X 1 G 1 + 2 X 2 G 2 32 X 1 G 2   2 + 32 G 2 3 + 3 G 2   2 2 G 2 − 16 G 1 G 2 2   X 1 X 2 + 4 G 1 G 2 − 8 G 2 3 X 2     2 + 32 G 2 3 + 3 G 2  2 G 2 − 28 G 1 G 2 2 6 G 1  8

Main computer algebra building block : polynomial elimination Polynomial elimination: ◮ Given an ideal I ⊂ K [ X 1 , . . . , X n , G 1 , . . . , G r ] ◮ Compute a basis of I G = I ∩ K [ G 1 , . . . , G r ] Computing eliminations allows to... ◮ ... compute projections of varieties ◮ ... solve if finitely many solutions (by iterating) ◮ ... compute unions and differences of varieties (by lifing) Many tools: resultants, triangular sets, Gröbner bases  2 X 1 X 2 + 4 G 1 G 2 − 8 G 2      2 G 1 − 3 X 2 2 G 2 − 3 G 2 2  2 X 1 X 1 G 1 + 2 X 2 G 2        3 + 3 X 1 G 2 2 − 12 X 2 G 1 G 2 2 + 56 X 2 G 2 3 X 1 G 1 + 2 X 2 G 2 32 X 1 G 2   2 + 32 G 2 3 + 3 G 2   2 2 G 2 − 16 G 1 G 2 2   X 1 X 2 + 4 G 1 G 2 − 8 G 2 3 X 2     2 + 32 G 2 3 + 3 G 2  2 G 2 − 28 G 1 G 2 2 6 G 1  8

Other previous works Non-commutative case Generic complexity Gröbner bases and strategy questions GB over rings Signature GB Polynomial Structured Extended systems MRI, control Power series Determinantal Valuations Critical points Rectangular matrices Complexity bounds Other classifications Real geometry 9

Other previous works Non-commutative case Generic complexity Gröbner bases and strategy questions GB over rings Signature GB Crypto., physics... Weighted homogeneous More general weights Automatic detection Polynomial Structured Extended systems MRI, control Power series Determinantal Valuations Critical points Rectangular matrices Complexity bounds Other classifications Real geometry 9

Other previous works Non-commutative case Generic complexity Gröbner bases and strategy questions GB over rings Number theory Signature GB Crypto., physics... Gröbner bases on Z Weighted homogeneous More criteria More general weights Beter implementation Automatic detection Polynomial Structured Extended systems MRI, control Power series Determinantal Valuations Critical points Rectangular matrices Complexity bounds Other classifications Real geometry 9

Other previous works Non-commutative case Generic complexity Gröbner bases and strategy questions GB over rings Number theory Signature GB Crypto., physics... Gröbner bases on Z Algebraic geometry Weighted homogeneous More criteria Gröbner bases More general weights Beter implementation on Tate algebras Automatic detection Faster reductions Polynomial Structured Extended systems MRI, control Power series Determinantal Valuations Critical points Rectangular matrices Complexity bounds Other classifications Real geometry 9