Computing the real solutions of polynomial systems with the - PowerPoint PPT Presentation

Computing the real solutions of polynomial systems with the RegularChains library in Maple Presented by Marc Moreno Maza 1 joint work with Changbo Chen 1 , James H. Davenport 2 , Fran¸ cois Lemaire 3 , Bican Xia 4 , Rong Xiao 1 Yuzhen Xie 1 1 University of Western Ontario 2 University of Bath (England) 3 Universit´ e de Lille 1 (France) 4 Peking University (China) ISSAC 2011 Software Presentation San Jose CA, June 9, 2011 (CDMMXX) RealTriangularize ISSAC 2011 1 / 17

Plan Overview 1 Solver verification 2 Branch cut analysis 3 Biochemical network analysis 4 Reachibility problem for hybrid systems 5 (CDMMXX) RealTriangularize ISSAC 2011 2 / 17

Overview Plan Overview 1 Solver verification 2 Branch cut analysis 3 Biochemical network analysis 4 Reachibility problem for hybrid systems 5 (CDMMXX) RealTriangularize ISSAC 2011 3 / 17

Overview The RegularChains library in Maple Design goals Solving polynomial systems over Q and F p , including parametric systems and semi-algebraic systems. Offering tools to manipulate their solutions. Organized around the concept of a regular chain , accommodating all types of solving and providing space-and-time efficiency. Features Use of types for algebraic structures: polynomial ring , regular chain , constructible set , quantifier free formula , regular semi algebraic system . Top level commands: PolynomialRing , Triangularize , RealTriangularize SamplePoints , . . . Tool kits: ConstructibleSetTools , ParametricSystemTools , FastArithmeticTools , SemiAlgebraicSetTools , . . . (CDMMXX) RealTriangularize ISSAC 2011 4 / 17

Overview The RegularChains library in Maple Design goals Solving polynomial systems over Q and F p , including parametric systems and semi-algebraic systems. Offering tools to manipulate their solutions. Organized around the concept of a regular chain , accommodating all types of solving and providing space-and-time efficiency. Features Use of types for algebraic structures: polynomial ring , regular chain , constructible set , quantifier free formula , regular semi algebraic system . Top level commands: PolynomialRing , Triangularize , RealTriangularize SamplePoints , . . . Tool kits: ConstructibleSetTools , ParametricSystemTools , FastArithmeticTools , SemiAlgebraicSetTools , . . . (CDMMXX) RealTriangularize ISSAC 2011 4 / 17

Overview Solving for the real solutions of polynomial systems Classical tools Isolating the real solutions of zero-dimensional polynomial systems: SemiAlgebraicSetTools:-RealRootIsolate Real root classification of parametric polynomial systems: ParametricSystemTools:-RealRootClassification Cylindrical algebraic decomposition of polynomial systems: SemiAlgebraicSetTools:-CylindricalAlgebraicDecompose New tools Triangular decomposition of semi-algebraic systems: RealTriangularize Sampling all connected components of a semi-algebraic system: SamplePoints Set-theoretical operations on semi-algebraic sets: SemiAlgebraicSetTools:-Difference (CDMMXX) RealTriangularize ISSAC 2011 5 / 17

Overview Solving for the real solutions of polynomial systems Classical tools Isolating the real solutions of zero-dimensional polynomial systems: SemiAlgebraicSetTools:-RealRootIsolate Real root classification of parametric polynomial systems: ParametricSystemTools:-RealRootClassification Cylindrical algebraic decomposition of polynomial systems: SemiAlgebraicSetTools:-CylindricalAlgebraicDecompose New tools Triangular decomposition of semi-algebraic systems: RealTriangularize Sampling all connected components of a semi-algebraic system: SamplePoints Set-theoretical operations on semi-algebraic sets: SemiAlgebraicSetTools:-Difference (CDMMXX) RealTriangularize ISSAC 2011 5 / 17

Overview Regular semi-algebraic system Notation Let T ⊂ Q [ x 1 < . . . < x n ] be a regular chain with y := { mvar ( t ) | t ∈ T } and u := x \ y = u 1 , . . . , u d . Let P be a finite set of polynomials, s.t. every f ∈ P is regular modulo sat ( T ). Let Q be a quantifier-free formula of Q [ u ]. Definition We say that R := [ Q , T , P > ] is a regular semi-algebraic system if: ( i ) Q defines a non-empty open semi-algebra ic set S in R d , ( ii ) the regular system [ T , P ] specializes well at every point u of S ( iii ) at each point u of S , the specialized system [ T ( u ) , P ( u ) > ] has at least one real solution. Z R ( R ) = { ( u , y ) | Q ( u ) , t ( u , y ) = 0 , p ( u , y ) > 0 , ∀ ( t , p ) ∈ T × P } . (CDMMXX) RealTriangularize ISSAC 2011 6 / 17

Overview Regular semi-algebraic system Notation Let T ⊂ Q [ x 1 < . . . < x n ] be a regular chain with y := { mvar ( t ) | t ∈ T } and u := x \ y = u 1 , . . . , u d . Let P be a finite set of polynomials, s.t. every f ∈ P is regular modulo sat ( T ). Let Q be a quantifier-free formula of Q [ u ]. Definition We say that R := [ Q , T , P > ] is a regular semi-algebraic system if: ( i ) Q defines a non-empty open semi-algebra ic set S in R d , ( ii ) the regular system [ T , P ] specializes well at every point u of S ( iii ) at each point u of S , the specialized system [ T ( u ) , P ( u ) > ] has at least one real solution. Z R ( R ) = { ( u , y ) | Q ( u ) , t ( u , y ) = 0 , p ( u , y ) > 0 , ∀ ( t , p ) ∈ T × P } . (CDMMXX) RealTriangularize ISSAC 2011 6 / 17

Overview Example The system [ Q , T , P > ], where � y 2 − a = 0 Q := a > 0 , T := , P > := { y > 0 } x = 0 is a regular semi-algebraic system. (CDMMXX) RealTriangularize ISSAC 2011 7 / 17

Overview RealTriangularize applied to the Eve surface (1/2) (CDMMXX) RealTriangularize ISSAC 2011 8 / 17

Overview RealTriangularize applied to the Eve surface (2/2) (CDMMXX) RealTriangularize ISSAC 2011 9 / 17

Solver verification Plan Overview 1 Solver verification 2 Branch cut analysis 3 Biochemical network analysis 4 Reachibility problem for hybrid systems 5 (CDMMXX) RealTriangularize ISSAC 2011 10 / 17

Solver verification Are these two different output equivalent? Given a triangle with edge lengths a , b , c (denoting the respective edges a , b , c too) the following two conditions S 1 , S 2 are both characterizing the fact that the external bi- sector of the angle of a , c intersects with b on the other side of a than the triangle: S 1 = a > 0 ∧ b > 0 ∧ c > 0 ∧ a < b + c ∧ b < a + c ∧ c < a + b ∧ b 2 + a 2 − c 2 ≤ 0 c ( b 2 + a 2 − c 2 ) 2 < ab 2 (2 ac − ( c 2 + a 2 − b 2 )) � � � � ∨ , S 2 = a > 0 ∧ b > 0 ∧ c > 0 ∧ a < b + c ∧ b < a + c ∧ c < a + b ∧ c − a > 0 . (CDMMXX) RealTriangularize ISSAC 2011 11 / 17

Branch cut analysis Plan Overview 1 Solver verification 2 Branch cut analysis 3 Biochemical network analysis 4 Reachibility problem for hybrid systems 5 (CDMMXX) RealTriangularize ISSAC 2011 12 / 17

Branch cut analysis Is this simplification correct? The original problem The branch cut of √ z is conventionally: { z ∈ C | ℜ ( z ) < 0 ∧ ℑ ( z ) = 0 } . Do the following equations hold for all z ∈ C : √ z − 1 √ z + 1 = √ z 2 − 1 and √ 1 − z √ 1 + z = √ 1 − z 2 . Turning the question to sampling The branch cuts of each formula is a semi-algebraic system S given as the disjunction of 3 others S 1 , S 2 , S 3 (one per √ ). Consider CAD-cells C 1 , . . . , C e , forming an intersection-free basis refining the connected components of S 1 , S 2 , S 3 . By virtue of the Modromy Theorem , it is sufficient to check whether the formula holds at a sample point of each of C 1 , . . . , C e . (CDMMXX) RealTriangularize ISSAC 2011 13 / 17

Branch cut analysis Is this simplification correct? The original problem The branch cut of √ z is conventionally: { z ∈ C | ℜ ( z ) < 0 ∧ ℑ ( z ) = 0 } . Do the following equations hold for all z ∈ C : √ z − 1 √ z + 1 = √ z 2 − 1 and √ 1 − z √ 1 + z = √ 1 − z 2 . Turning the question to sampling The branch cuts of each formula is a semi-algebraic system S given as the disjunction of 3 others S 1 , S 2 , S 3 (one per √ ). Consider CAD-cells C 1 , . . . , C e , forming an intersection-free basis refining the connected components of S 1 , S 2 , S 3 . By virtue of the Modromy Theorem , it is sufficient to check whether the formula holds at a sample point of each of C 1 , . . . , C e . (CDMMXX) RealTriangularize ISSAC 2011 13 / 17

Biochemical network analysis Plan Overview 1 Solver verification 2 Branch cut analysis 3 Biochemical network analysis 4 Reachibility problem for hybrid systems 5 (CDMMXX) RealTriangularize ISSAC 2011 14 / 17

Biochemical network analysis Is there a unique positive equilibrium? Allosteric enzym Cascad of polymerisation k 1 − ⇀ k 2 E + S ↽ − C − − ⇀ P 1 + P 1 − − P 2 ↽ k 2 k − 2 k 3 − ⇀ E + C − F ↽ k 3 − − ⇀ P 1 + P 2 − − P 3 ↽ k 4 k − 3 2 C − 1 1 2 E + S − 1 2 C 0 + 1 2 E 0 − S 0 = 0 . . . 1 2 C + 1 2 E + F − 1 2 C 0 − 1 = 0 2 E 0 − F 0 k 1 ES − k 2 C − k 3 EC + k 4 F = 0 k n − − ⇀ P 1 + P n − 1 ↽ − − P n − 2 k 3 EC + 2 k 4 F = 0 k − E , S , C , F , E 0 , S 0 , C 0 , F 0 , k 1 , k 2 , k 3 , k 4 0 . > n Each system is viewed as parametric in the initial concentrations and kinetic velocities. We show that, generically, there is a unique positive equilibrium. (CDMMXX) RealTriangularize ISSAC 2011 15 / 17