Exact Computation of the Real Solutions of Arbitrary Polynomial Systems

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1 Exact Computation of the Real Solutions of Arbitrary Polynomial Systems Presented by Marc Moreno Maza 1 joint work with Changbo Chen 1, James H. Davenport 2, François Lemaire 3, John P. May 5, Bican Xia 4, Rong Xiao 1 and Yuzhen Xie 1 1 University of Western Ontario (Canada) 2 University of Bath (England) 3 Université de Lille 1 (France) 4 Peking University (China) 5 Maplesoft Inc. (Canada) ICIAM 2011 Computer Algebra Mini-symposium Vancouver BC, July 18, 2011 (CDMMXX) RealTriangularize ICIAM / 32

2 Plan 1 Computing the real solutions of polynomial systems symbolically 2 Triangular decomposition of semi-algebraic systems 3 Application to dynamical system analysis (CDMMXX) RealTriangularize ICIAM / 32

3 Plan 1 Computing the real solutions of polynomial systems symbolically 2 Triangular decomposition of semi-algebraic systems 3 Application to dynamical system analysis (CDMMXX) RealTriangularize ICIAM / 32

4 Solving polynomial systems? What does this mean? The algebra text book says: For F k[x 1,..., x n ] this is simply a primary decomposition of F or the irreducible decomposition of V (F ) (the zero set of F in k n ). The computer algebra system does well: For F k[x 1,..., x n ], with k = Z/pZ or k = Q, computing a Gröbner basis of F or computing a triangular decomposition of V (F ). But most scientists and engineers need: For F Q[x 1,..., x n ], a useful description of the points of V (F ) whose coordinates are real. For F Q[u 1,..., u d ][x 1,..., x n ], the real (x 1,..., x n )-solutions as a function of the real parameter (u 1,..., u d ). (CDMMXX) RealTriangularize ICIAM / 32

5 Solving polynomial systems? What does this mean? The algebra text book says: For F k[x 1,..., x n ] this is simply a primary decomposition of F or the irreducible decomposition of V (F ) (the zero set of F in k n ). The computer algebra system does well: For F k[x 1,..., x n ], with k = Z/pZ or k = Q, computing a Gröbner basis of F or computing a triangular decomposition of V (F ). But most scientists and engineers need: For F Q[x 1,..., x n ], a useful description of the points of V (F ) whose coordinates are real. For F Q[u 1,..., u d ][x 1,..., x n ], the real (x 1,..., x n )-solutions as a function of the real parameter (u 1,..., u d ). (CDMMXX) RealTriangularize ICIAM / 32

6 Solving polynomial systems? What does this mean? The algebra text book says: For F k[x 1,..., x n ] this is simply a primary decomposition of F or the irreducible decomposition of V (F ) (the zero set of F in k n ). The computer algebra system does well: For F k[x 1,..., x n ], with k = Z/pZ or k = Q, computing a Gröbner basis of F or computing a triangular decomposition of V (F ). But most scientists and engineers need: For F Q[x 1,..., x n ], a useful description of the points of V (F ) whose coordinates are real. For F Q[u 1,..., u d ][x 1,..., x n ], the real (x 1,..., x n )-solutions as a function of the real parameter (u 1,..., u d ). (CDMMXX) RealTriangularize ICIAM / 32

7 Solving for the real solutions: classical techniques In dimension zero over Q: For F Q[x 1,..., x n ], if V (F ) is finite, many standard and efficient techniques apply to identify the real solutions. In (generic) dimension zero over Q[u 1,..., u d ]: For F Q[u 1,..., u d ][x 1,..., x n ] and an integer r one can determine generic conditions on u 1,..., u d for F to admit exactly r real (x 1,..., x n )-solutions For arbitrary systems: For F Q[x 1,..., x n ], one can partition R n into cylindrical cells where the sign of each f F does not change. (CDMMXX) RealTriangularize ICIAM / 32

8 Solving for the real solutions: classical techniques In dimension zero over Q: For F Q[x 1,..., x n ], if V (F ) is finite, many standard and efficient techniques apply to identify the real solutions. In (generic) dimension zero over Q[u 1,..., u d ]: For F Q[u 1,..., u d ][x 1,..., x n ] and an integer r one can determine generic conditions on u 1,..., u d for F to admit exactly r real (x 1,..., x n )-solutions For arbitrary systems: For F Q[x 1,..., x n ], one can partition R n into cylindrical cells where the sign of each f F does not change. (CDMMXX) RealTriangularize ICIAM / 32

9 Solving for the real solutions: classical techniques In dimension zero over Q: For F Q[x 1,..., x n ], if V (F ) is finite, many standard and efficient techniques apply to identify the real solutions. In (generic) dimension zero over Q[u 1,..., u d ]: For F Q[u 1,..., u d ][x 1,..., x n ] and an integer r one can determine generic conditions on u 1,..., u d for F to admit exactly r real (x 1,..., x n )-solutions For arbitrary systems: For F Q[x 1,..., x n ], one can partition R n into cylindrical cells where the sign of each f F does not change. (CDMMXX) RealTriangularize ICIAM / 32

10 Real root isolation for zero-dimensional systems (CDMMXX) RealTriangularize ICIAM / 32

11 Real root classification: generically 0-dimensional systems (CDMMXX) RealTriangularize ICIAM / 32

12 Cylindrical algebraic decomposition of {ax 2 + bx + c} root a = 0 a <, > 0 b = 0 b <, > 0 C c = 0 c <, > 0 C r = 0 r <, > 0 C C t = 0 t <, > 0 q = 0 q <, > 0p = 0 p <, > 0 The cylindrical algebraic decomposition of {ax 2 + bx + c} is given by the tree above, where t = bx + c, q = 2ax + b, and r = 4ac b 2. This is the best possible output for that method, leading to 27 cells! (CDMMXX) RealTriangularize ICIAM / 32

13 Can a computer program be as good as a high-school student? For the equation ax 2 + bx + c = 0, can a computer program produce? 2ax + b = 0 ax 2 + bx + c = 0 4ac b 2 = 0 a 0 b 2 4ac > 0 a 0 bx + c = 0 a = 0 b 0 c = 0 b = 0 a = 0 (CDMMXX) RealTriangularize ICIAM / 32

14 Yes, our new algorithm RealTriangularize can do that! (CDMMXX) RealTriangularize ICIAM / 32

15 RealTriangularize applied to the Eve surface (1/2) (CDMMXX) RealTriangularize ICIAM / 32

16 RealTriangularize applied to the Eve surface (2/2) (CDMMXX) RealTriangularize ICIAM / 32

17 Plan 1 Computing the real solutions of polynomial systems symbolically 2 Triangular decomposition of semi-algebraic systems 3 Application to dynamical system analysis (CDMMXX) RealTriangularize ICIAM / 32

18 Regular chain Example f 3 = x 4 x x T := f 2 = (x 1 + x 2 )x x f 1 = x Under the order x 5 > x 4 > x 3 > x 2 > x 1, mvar(f 2 ) = x 3 and mvar(f 1 ) = x 1 init(f 2 ) = x 1 + x 2 and init(f 1 ) = 1 T is a regular chain. Indeed: Proposition init(f2 ) is regular (neither zero nor zero-divisor) modulo f 1. init(f3 ) is regular modulo f 1, f 2 : init(f 2 ). Let T = {t 1,..., t s } be a regular chain of k[x]. Then the saturated ideal sat(t ) := T : ( s i=1 init(t i)) is a proper equi-dimensional ideal of k[x]. (CDMMXX) RealTriangularize ICIAM / 32

19 Regular chain Example f 3 = x 4 x x T := f 2 = (x 1 + x 2 )x x f 1 = x Under the order x 5 > x 4 > x 3 > x 2 > x 1, mvar(f 2 ) = x 3 and mvar(f 1 ) = x 1 init(f 2 ) = x 1 + x 2 and init(f 1 ) = 1 T is a regular chain. Indeed: Proposition init(f2 ) is regular (neither zero nor zero-divisor) modulo f 1. init(f3 ) is regular modulo f 1, f 2 : init(f 2 ). Let T = {t 1,..., t s } be a regular chain of k[x]. Then the saturated ideal sat(t ) := T : ( s i=1 init(t i)) is a proper equi-dimensional ideal of k[x]. (CDMMXX) RealTriangularize ICIAM / 32

20 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 ICIAM / 32

21 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 ICIAM / 32

22 Example The system [Q, T, P > ], where { y Q := a > 0, T := 2 a = 0 x = 0, P > := {y > 0} is a regular semi-algebraic system. (CDMMXX) RealTriangularize ICIAM / 32

23 RealTriangularize applied to sofa and cylinder (1/2) x 2 + y 3 + z 5 = x 4 + z 2 1 = 0 (CDMMXX) RealTriangularize ICIAM / 32

24 RealTriangularize applied to sofa and cylinder (2/2) (CDMMXX) RealTriangularize ICIAM / 32

25 The RegularChains library in Maple (1/2) 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 ICIAM / 32

26 The RegularChains library in Maple (1/2) 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 ICIAM / 32

27 The RegularChains library in Maple (2/2) 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 ICIAM / 32

28 The RegularChains library in Maple (2/2) 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 ICIAM / 32

29 Plan 1 Computing the real solutions of polynomial systems symbolically 2 Triangular decomposition of semi-algebraic systems 3 Application to dynamical system analysis (CDMMXX) RealTriangularize ICIAM / 32

30 A biochemical network: setting (1/3) The generic kinetic scheme of prion diseases is illustrated as follows: 1 PrP C 3 PrP S C 4 Aggregates. 2 where [PrP C ] is the concentration of PrP C (harmless form) [ PrP S C ] is the concentration of PrP S C (infectious form) which catalyses the transformation from the normal form to itself, Step 1: synthesis of native PrP C Step 2, 4: normal degradation. (CDMMXX) RealTriangularize ICIAM / 32

31 A biochemical network: setting (2/3) Let ν i be the rate of Step i for i = 1,..., 4. 1 PrP C 3 PrP S C 4 Aggregates. 2 Step 1: zero-order kinetic process, that is ν 1 = k 1, Step 2, 4: first-order rate equations: ν 2 = k 2 [ PrP C ], ν 4 = k 4 [ PrP S C ]. Step 3: a nonlinear process ν 3 = [PrP ] C a ( 1 + b [ PrP S ] C n ) 1 + c [PrP S n. C ] (CDMMXX) RealTriangularize ICIAM / 32

32 A biochemical network: setting (3/3) We also have: 1 PrP C 3 PrP S C 4 Aggregates. 2 d [ PrP C ] = ν 1 ν 2 ν 3 dt d [ PrP S ] C = ν 3 ν 4 dt Letting x = [ PrP C ] and y = [ PrP S C ]. we obtain the dynamical system: dx dt = k 1 k 2 x ax (1 + by n ) 1 + cy n dy dt = ax (1 + by n ) 1 + cy n k 4y (CDMMXX) RealTriangularize ICIAM / 32

33 A biochemical network: semi-algebraic systems to solve Dynamical system to study M. Laurent (Biochem. J., 1996) suggests to set b = 2, c = 1/20, n = 4, a = 1/10, k 4 = 50 and k 1 = 800, leading to: { dx dt = f 1 dy dt = f 2 with { f1 = y 4 20k 2 x k 2 xy 4 2x 4xy 4 20+y 4 f 2 = 2(x+2xy 4 500y 25y 5 ) 20+y 4. Semi-algebraic systems to solve By Routh-Hurwitz criterion, an equilibrium (x, y) is asymptotically stable if 1 := ( f 1 x + f 2 y ) > 0 and a 2 := f 1 x f 2 y f 1 y f 2 x > 0. Letting p 1, p 2 the above polynomials, we obtain two semi-algebraic systems: S 1 : {p 1 = p 2 = 0, k 2 > 0} and S 2 : {p 1 = p 2 = 0, k 2 > 0, 1 > 0, a 2 > 0} (CDMMXX) RealTriangularize ICIAM / 32

34 A biochemical network: solving S 1 The real solutions of S 1 are described by the following triangular decomposition into regular semi-algebraic systems. (2y 4 + 1)x 25y 5 500y = 0 (k A 1 := 2 + 4)y 5 64y 4 + (2 + 20k 2 )y 32 = 0 k 2 > 0 r 0, A 2 := t x = 0 t y = 0 r = 0 k 2 > 0. where t y (k 2, y) has degree 4 in y and r is given by r := k k k k k k k k The polynomial r has four real roots, two are positive: α 1 < α 2. (CDMMXX) RealTriangularize ICIAM / 32

35 A biochemical network: solving S 1 Regarding k 2 as a parameter, one can compute a real comprehensive triangular decomposition which gives: { } k 2 0 {A 1 } 0 < k 2 < α 1 {A 2 } k 2 = α 1 {A 1 } α 1 < k 2 < α 2 {A 2 } k 2 = α 2 {A 1 } k 2 > α 2 From where we deduce the number of real solutions: 0 k k 2 > 0 and r > 0 2 k 2 > 0 and r = 0 3 k 2 > 0 and r < 0. (CDMMXX) RealTriangularize ICIAM / 32

36 A biochemical network: conclusion Theorem Assume that k 2 > 0. Then we have: if r > 0, then the dynamical system has 1 equilibrium; if r = 0, then it has 2 equilibria; if r < 0, it has 3 equilibria. Theorem Assume that k 2 > 0. Then we have: if r > 0, then the system has one hyperbolic equilibrium, which is asymptotically stable; if r < 0 and r 2 0, then the system has three hyperbolic equilibria, two of which are asymptotically stable and the other one is unstable; if r = 0 or r 2 = 0, the system experiences a bifurcation where r 2 = k k k k k k k k k (CDMMXX) RealTriangularize ICIAM / 32

37 (CDMMXX) RealTriangularize ICIAM / 32

38 (CDMMXX) RealTriangularize ICIAM / 32

39 (CDMMXX) RealTriangularize ICIAM / 32

40 Summary and notes Solving for the real roots of (parametric or not) polynomial systems is a fundamental problem with many applications. Most of the time, this requires exact (thus symbolic) computation. Computer algebra systems used to have limited capabilities for that, especially for the parametric case. Recent work (Changbo Chen, James H. Davenport, M 3, Bican Xia & Rong Xiao, ISSAC ) is changing that. RealTriangularize is available Maple 15, as part of the Maple RegularChains library. (CDMMXX) RealTriangularize ICIAM / 32

Computing the real solutions of polynomial systems with the RegularChains library in Maple

ISSAC 2011 Software Presentation San Jose CA, June 9, 2011 (CDMMXX) RealTriangularize ISSAC 2011 1 / 17 Computing the real solutions of polynomial systems with the RegularChains library in Maple Presented