Sage for Dynamical Systems

Transcription

1 for Dynamical Systems Benjamin Hutz Department of Mathematics and Computer Science Saint Louis University December 5, 2015 RTG Workshop: Arithmetic Dynamics This work is supported by NSF grant DMS Benjamin Hutz for Dynamical Systems 1 / 48

2 What is? Math is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Access their combined power through a common, Python-based language or directly via interfaces or wrappers. Mission Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab. Benjamin Hutz for Dynamical Systems 2 / 48

3 How can you use Online Locally Download from Prebuilt binaries Install from source code Benjamin Hutz for Dynamical Systems 3 / 48

4 Dynamical Systems and -dynamics project 1 Started at ICERM in days 55 in NSF grant DMS , (PI: Hutz) 4 Future -days?2017/2018? Resources Project Page: dynamics/arithmeticandcomplex Google group: sage-dynamics Benjamin Hutz for Dynamical Systems 4 / 48

5 Basic Functionality 1 Projective space, affine space, and subschemes (varieties) 2 Points and functions 3 Iteration 4 Heights and canonical heights 5 Periodic points, dynatomic polynomials 6 Critical points Benjamin Hutz for Dynamical Systems 5 / 48

6 Basic Examples: Iteration 1 sage: A.<x> = AffineSpace(QQ,1) 2 sage: f = DynamicalSystem_affine([xˆ2-1], domain=a) 3 sage: P=A(2) 4 sage: f(p), f(f(p)), f(f(f(p))), f.nth_iterate(p,4) 5 ((3), (8), (63), (3968)) 6 sage: f.orbit(p,[0,3]) 7 [(2), (3), (8), (63)] 8 sage: f.nth_iterate_map(2) 9 Scheme endomorphism of Affine Space of dimension 1 over Rational Field 10 Defn: Defined on coordinates by sending (x) to 11 (xˆ4-2*xˆ2) Benjamin Hutz for Dynamical Systems 6 / 48

7 Basic Examples: Varieties 1 sage: A.<x,y> = AffineSpace(QQ,2) 2 sage: X = A.subscheme([x-1, yˆ2-4]); 3 sage: X.dimension() sage: X.rational_points() 6 [(1, -2), (1, 2)] 7 sage: X.defining_ideal() 8 Ideal (x - 1, yˆ2-4) of Multivariate Polynomial Ring in x, y over Rational Field 9 sage: X.coordinate_ring() 10 Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x - 1, yˆ2-4) Benjamin Hutz for Dynamical Systems 7 / 48

8 Height definitions Given a point P = (P 0 : : P N ) The (absolute) global height is defined as h(p) = 1 [K : Q] log(max( P i )) The local height (Green s function) at a place v is defined as λ v (P) = log(max( P i v )) The canonical height with respect to a morphism f is defined as h(f n (P)) ĥ(p) = lim n deg(f ) n. Benjamin Hutz for Dynamical Systems 8 / 48

9 Basic Examples: Heights 1 sage: P.<x,y> = ProjectiveSpace(QQ,1) 2 sage: f = DynamicalSystem([4*xˆ2-3*yˆ2,4*yˆ2], domain=p) 3 sage: Q = P(3,5) 4 sage: f.green_function(q,2, prec=100, N=20) sage: Q.global_height() sage: Q.canonical_height(f,error_bound=0.001) Benjamin Hutz for Dynamical Systems 9 / 48

10 Basic Examples: Local Heights 1 sage: K.<w> = QuadraticField(3) 2 sage: P.<x,y> = ProjectiveSpace(K,1) 3 sage: f = DynamicalSystem([17*xˆ2+1/7*yˆ2,17*w*x*y], domain=p) 4 sage: f.green_function(p([2,1]), K.ideal(7), N=7) sage: f.green_function(p([w,1]), K.ideal(17), error_bound=0.001) Benjamin Hutz for Dynamical Systems 10 / 48

11 Definitions: Dynatomic Polynomial Let f K [z] be a single variable polynomial. Definition The n-th dynatomic polynomial for f is defined as Φ n(f ) = d n(f d (z) z) µ(n/d). Definition We can also define a generalized dynatomic polynomial for preperiodic points as Φ m,n(f ) = Φ n(f m ) Φ n(f m 1 ). More general definitions can be made using intersection theory. Benjamin Hutz for Dynamical Systems 11 / 48

12 Basic Examples: Dynatomic Polynomial 1 sage: R.<c> = FunctionField(QQ) 2 sage: P.<x,y> = ProjectiveSpace(R,1) 3 sage: f = DynamicalSystem([xˆ2+c*yˆ2,yˆ2], domain=p ) 4 sage: f.dynatomic_polynomial(3) 5 xˆ6 + xˆ5*y + (3*c + 1)*xˆ4*yˆ2 + (2*c + 1)*xˆ3*yˆ3 + (3*cˆ2 + 3*c + 1)*xˆ2*yˆ4 + (cˆ2 + 2*c + 1)* x*yˆ5 + (cˆ3 + 2*cˆ2 + c + 1)*yˆ6 6 sage: f.dynatomic_polynomial(3).subs({x:0,y:1}) 7 cˆ3 + 2*cˆ2 + c sage: f.nth_iterate(p(0),3) 9 (cˆ4 + 2*cˆ3 + cˆ2 + c : 1) Benjamin Hutz for Dynamical Systems 12 / 48

13 Basic Examples: Number Fields 1 sage: A.<z> = AffineSpace(QQ,1) 2 sage: f = DynamicalSystem([zˆ2+1], domain=a) 3 sage: R = A.coordinate_ring() 4 sage: K.<w> = NumberField(f.dynatomic_polynomial(2).polynomial(R.gen(0))) 5 sage: A.<z> = AffineSpace(K,1) 6 sage: f = DynamicalSystem_affine([zˆ2+1]) 7 sage: f.orbit(a(w),3) 8 [(w), (-w - 1), (w), (-w - 1)] Benjamin Hutz for Dynamical Systems 13 / 48

14 Basic Examples: All Periodic Points 1 sage: P.<x,y> = ProjectiveSpace(QQbar,1) 2 sage: H = End(P) 3 sage: f = H([xˆ2+yˆ2,yˆ2]) 4 sage: f.periodic_points(2) 5 [( ? ?*I : 1), 6 ( ? ?*I : 1)] Benjamin Hutz for Dynamical Systems 14 / 48

15 Basic Examples: Critical Points 1 sage: P.<x,y> = ProjectiveSpace(QQ,1) 2 sage: f = DynamicalSystem([xˆ2+yˆ2,yˆ2]) 3 sage: f.critical_points() 4 [(0 : 1), (1 : 0)] 5 sage: f.critical_height(error_bound=0.001) Benjamin Hutz for Dynamical Systems 15 / 48

16 Basic Examples: Finite Fields 1 sage: P.<x,y> = ProjectiveSpace(GF(13),1) 2 sage: f = DynamicalSystem([xˆ2+yˆ2,yˆ2]) 3 sage: f.cyclegraph() 4 Looped digraph on 14 vertices 5 sage: f.orbit_structure(p([2,1])) 6 [0, 4] Benjamin Hutz for Dynamical Systems 16 / 48

17 Rational preperiodic points Post-critically finite maps Automorphism Groups 1 Rational preperiodic points 1 Determine all of the rational preperiodic points for a morphism f : P N P N. 2 Determine if a given point is preperiodic for f. 2 Post-critically finite maps 1 Determine if a morphism f : P 1 P 1 is post-critically finite 2 Construct a pcf map with specified portrait 3 Automorphism groups 1 Construct a family of maps each of whose automorphism group contains the given finite group. 4 1 Products of projective spaces and Wehler K3 surfaces 2 Iteration of subvarieties Benjamin Hutz for Dynamical Systems 17 / 48

18 Rational preperiodic points Post-critically finite maps Automorphism Groups Definitions Theorem (Northcott) The set of rational preperiodic points for a morphism f : P N P N is a set of bounded height. Conjecture (Morton-Silverman) Given a morphism f : P N P N of degree d, defined over a number field of degree D, then there exists a constant C(d, D, N) such that # PrePer(f ) C(d, D, N). Benjamin Hutz for Dynamical Systems 18 / 48

19 Rational preperiodic points Post-critically finite maps Automorphism Groups A precise description of n Theorem (Morton-Silverman 1994, Zieve) Let f : P 1 P 1 defined over Q with deg(f ) 2. Assume that f has good reduction at p with a rational periodic point P. Define Then n = minimal period of P. m = minimal period of P modulo p. r = the multiplicative order of (f m ) (P) mod p n = m or n = mrp e for some explicitly bounded integer e 0. Benjamin Hutz for Dynamical Systems 19 / 48

20 Rational preperiodic points Post-critically finite maps Automorphism Groups Algorithm for finding all rational preperiodic points [Hut15] 1 For each prime p in PrimeSet with good reduction find the list of possible global periods: 1 Find all of the periodic cycles modulo p 2 Compute m, mrp e for each cycle. 2 Intersect the lists of possible periods for all primes in PrimeSet. 3 For each n in PossiblePeriods 1 Find all rational solutions to f n (x) = x. 4 Let PrePeriodicPoints = PeriodicPoints. 5 Repeat until PrePeriodicPoints is constant 1 Add the first rational preimage of each point in PrePeriodicPoints to PrePeriodicPoints. Use Weil restriction of scalars for number fields (polynomials in dimension 1). Benjamin Hutz for Dynamical Systems 20 / 48

21 Rational preperiodic points Post-critically finite maps Automorphism Groups Examples 1 sage: P.<x,y> = ProjectiveSpace(QQ,1) 2 sage: f = DynamicalSystem([xˆ2-29/16*yˆ2,yˆ2]) 3 sage: f.rational_preperiodic_graph() 4 Looped digraph on 9 vertices 1 sage: R.<t> = PolynomialRing(QQ) 2 sage: K.<v> = NumberField(tˆ3 + 16*tˆ *t ) 3 sage: PS.<x,y> = ProjectiveSpace(K,1) 4 sage: f = DynamicalSystem([xˆ2-29/16*yˆ2,yˆ2]) # Hutz 5 sage: len(f.rational_preperiodic_points()) #6s 6 21 Benjamin Hutz for Dynamical Systems 21 / 48

22 Rational preperiodic points Post-critically finite maps Automorphism Groups Examples 1 sage: K.<w> = QuadraticField(33) 2 sage: PS.<x,y> = ProjectiveSpace(K,1) 3 sage: f = DynamicalSystem([xˆ2-71/48*yˆ2,yˆ2]) # Stoll 4 sage: len(f.rational_preperiodic_points()) #1s sage: P.<x,y,z> = ProjectiveSpace(QQ,2) 2 sage: f = DynamicalSystem([xˆ2-21/16*zˆ2,yˆ2-2*zˆ 2,zˆ2]) 3 sage: len(f.rational_preperiodic_points()) #1.5s 4 44 Benjamin Hutz for Dynamical Systems 22 / 48

23 Rational preperiodic points Post-critically finite maps Automorphism Groups Is a point preperiodic? Let f : P N P N be a morphism of degree d. Theorem A point P is preperiodic for f if and only if ĥ(p) = 0. Theorem There exists a constant C independent of a point P such that ĥ(p) h(p) C(f ). Theorem For all points P ĥ(f (P)) = dĥ(p). Benjamin Hutz for Dynamical Systems 23 / 48

24 Rational preperiodic points Post-critically finite maps Automorphism Groups Algorithm: is preperiodic() 1 If ĥ(p) > C, then not preperiodic. 2 Compute forward images 1 If we encounter a cycle, return preperiodic 2 If the height becomes > C, return not preperiodic. Benjamin Hutz for Dynamical Systems 24 / 48

25 Rational preperiodic points Post-critically finite maps Automorphism Groups Example 1 sage: P.<x,y,z> = ProjectiveSpace(QQ,2) 2 sage: f = DynamicalSystem([xˆ2-29/16*zˆ2,yˆ2-21/ 16*zˆ2,zˆ2]) 3 sage: Q = P(1,-3,4) 4 sage: Q.is_preperiodic(f,return_period=True) 5 (1,3) 1 sage: K.<w> = QuadraticField(-2) 2 sage: P.<x,y,z> = ProjectiveSpace(K,2) 3 sage: f = DynamicalSystem([xˆ2-29/16*zˆ2,yˆ2-21/ 16*zˆ2,zˆ2]) 4 sage: Q = P(w,-3,4) 5 sage: Q.is_preperiodic(f) 6 False Benjamin Hutz for Dynamical Systems 25 / 48

26 Rational preperiodic points Post-critically finite maps Automorphism Groups Definitions Definition We say that P is a critical point for f if the jacobian of f does not have maximal rank at P. Definition We say that f is post-critically finite if all of the critical points are preperiodic. Definition The critical height of f is defined as ĥ(c). c crit Benjamin Hutz for Dynamical Systems 26 / 48

27 Rational preperiodic points Post-critically finite maps Automorphism Groups Algorithm: is postcritically finite() 1 Compute the critical points of f over Q 2 Determine if each is preperiodic. Benjamin Hutz for Dynamical Systems 27 / 48

28 Rational preperiodic points Post-critically finite maps Automorphism Groups Example 1 sage: P.<x,y> = ProjectiveSpace(QQ,1) 2 sage: f = DynamicalSystem([xˆ2 + 12*yˆ2, 7*x*y]) 3 sage: f.critical_points(r=qqbar) 4 [( ? : 1), ( ? : 1 )] 5 sage: f.is_postcritically_finite() 6 False 7 sage: f.critical_height(error_bound=0.001) Benjamin Hutz for Dynamical Systems 28 / 48

29 Rational preperiodic points Post-critically finite maps Automorphism Groups Example: Lattès maps 1 sage: P.<x,y> = ProjectiveSpace(QQ,1) 2 sage: E = EllipticCurve([0,0,0,0,2]);E 3 Elliptic Curve defined by yˆ2 = xˆ3 + 2 over Rational Field 4 sage: f = P.Lattes_map(E,2) 5 sage: f.is_postcritically_finite() 6 True 7 sage: f.critical_point_portrait() 8 Looped digraph on 10 vertices 9 sage: f.critical_height(error_bound=0.001) e-17 Benjamin Hutz for Dynamical Systems 29 / 48

30 Rational preperiodic points Post-critically finite maps Automorphism Groups Misiurewicz Point 1 sage: R.<c> = PolynomialRing(QQ) 2 sage: A.<z> = AffineSpace(R,1) 3 sage: f = DynamicalSystem([zˆ2+c], domain=a) 4 sage: f.dynatomic_polynomial([2,2]).subs({z:0}) 5 cˆ2+1 1 sage: R.<x> = QQ[] 2 sage: K.<i> = NumberField(xˆ2+1) 3 sage: A.<z> = AffineSpace(K,1) 4 sage: f = DynamicalSystem([zˆ2+i], domain=a) 5 sage: f.orbit(a(0),4) 6 [(0), (i), (i - 1), (-i), (i - 1)] Benjamin Hutz for Dynamical Systems 30 / 48

31 Rational preperiodic points Post-critically finite maps Automorphism Groups Gleason Polynomial 1 sage: R.<c> = PolynomialRing(QQ) 2 sage: A.<z> = AffineSpace(R,1) 3 sage: f = DynamicalSystem([zˆ2+c], domain=a) 4 sage: K.<w> = NumberField(f.dynatomic_polynomial(3).subs({z:0})) 5 sage: A.<z> = AffineSpace(K,1) 6 sage: f = DynamicalSystem([zˆ2+w],domain=A) 7 sage: f.orbit(a(0),3) 8 [(0), (w), (wˆ2 + w), (0)] Benjamin Hutz for Dynamical Systems 31 / 48

32 Rational preperiodic points Post-critically finite maps Automorphism Groups 2-parameters [HT15]: generate ideal 1 sage: R.<a,v> = PolynomialRing(QQ,2, order='lex') 2 sage: P.<x,y> = ProjectiveSpace(R,1) 3 sage: f = DynamicalSystem([xˆ3-3*aˆ2*x*yˆ2 +(2*aˆ3 +v)*yˆ3,yˆ3]) 4 sage: dyn1 = f.dynatomic_polynomial(2) 5 sage: dyn2 = f.dynatomic_polynomial(1) 6 sage: I = R.ideal([dyn1.subs({x:a,y:1}), dyn2.subs( {x:-a,y:1})]) Benjamin Hutz for Dynamical Systems 32 / 48

33 Rational preperiodic points Post-critically finite maps Automorphism Groups 2-parameters: get solutions 1 sage: G = I.groebner_basis() 2 sage: S.<z> = QQ[] 3 sage: phi = R.hom([0,z],S) 4 sage: p = phi(g[-1]).factor() 5 sage: K.<V> = NumberField(p[0][0]) 6 sage: S.<z> = K[] 7 sage: phi2 = R.hom([z,V],S) 8 sage: q = phi2(g[-2]).factor() 9 sage: L.<A> = K.extension(q[0][0]) Benjamin Hutz for Dynamical Systems 33 / 48

34 Rational preperiodic points Post-critically finite maps Automorphism Groups 2-parameters: check orbits 1 sage: P.<x,y>=ProjectiveSpace(L,1) 2 sage: f=dynamicalsystem([xˆ3-3*aˆ2*x*yˆ2 +(2*Aˆ3+V )*yˆ3,yˆ3]) 3 sage: f.orbit(p(a),2) 4 [(2/3*Vˆ5 + Vˆ3 + 5/6*V : 1), (V : 1), (2/3*Vˆ5 + V ˆ3 + 5/6*V : 1)] 5 sage: f.orbit(p(-a),2) 6 [(-2/3*Vˆ5 - Vˆ3-5/6*V : 1), 7 (-2/3*Vˆ5 - Vˆ3-5/6*V : 1), 8 (-2/3*Vˆ5 - Vˆ3-5/6*V : 1)] Benjamin Hutz for Dynamical Systems 34 / 48

35 Rational preperiodic points Post-critically finite maps Automorphism Groups Definitions Definition Let f : P N P N. Then an element α PGL N+1 acts on f via conjugation f α = α f α 1 Definition The automorphism group of f is the group Aut(f ) = {α PGL N+1 : f α = f }. Benjamin Hutz for Dynamical Systems 35 / 48

36 Rational preperiodic points Post-critically finite maps Automorphism Groups Determining Automorphism Groups [FMV14] 1 sage: R.<x,y> = ProjectiveSpace(QQ,1) 2 sage: f = DynamicalSystem([xˆ2-2*x*y-2*yˆ2,-2*xˆ2-2 *x*y+yˆ2]) 3 sage: f.automorphism_group(return_functions=true) 4 [x, 2/(2*x), -x - 1, -2*x/(2*x + 2), (-x - 1)/x, -1 /(x + 1)] 5 sage: f.conjugate(matrix([[-1,-1],[0,1]])) == f 6 True Benjamin Hutz for Dynamical Systems 36 / 48

37 Rational preperiodic points Post-critically finite maps Automorphism Groups Constructing Automorphisms [dfh17] Definition Given a finite matrix group G, we say that a polynomial p is an invariant of G if Theorem (Klein) p g = p for all g G. If F is an invariant for a finite group G, then has G Aut(f ). f = ( F y : F x ) : P 1 P 1 Benjamin Hutz for Dynamical Systems 37 / 48

38 Rational preperiodic points Post-critically finite maps Automorphism Groups Tetrahedral family: Invariants 1 sage: R.<x> = QQ[] 2 sage: K.<i> = CyclotomicField(4) 3 sage: m1 = matrix(k,2,[(-1+i)/2,(-1+i)/2, (1+i)/2,( -1-i)/2]) 4 sage: m2 = matrix(k,2, [0,i, -i,0]) 5 sage: tetra = MatrixGroup(m1,m2) 6 sage: [p8,p12a,p12b] = tetra.invariant_generators() 7 sage: S = p8.parent() 8 sage: x1,x2 = S.gens() 9 sage: F7 = [-p8.derivative(x2),p8.derivative(x1)] 10 sage: F11 = [-p12a.derivative(x2),p12a.derivative( x1)] 11 sage: F5 = [-p12b.derivative(x2),p12b.derivative(x1 )] Benjamin Hutz for Dynamical Systems 38 / 48

39 Rational preperiodic points Post-critically finite maps Automorphism Groups Tetrahedral family: Morphisms 1 sage: P.<x,y> = ProjectiveSpace(K,1) 2 sage: R = P.coordinate_ring() 3 sage: phi = S.hom([x,y],R) 4 sage: f7 = DynamicalSystem([phi(t) for t in F7]) 5 sage: f11 = DynamicalSystem([phi(t) for t in F11]) 6 sage: f5 = DynamicalSystem([phi(t) for t in F5]) 7 sage: f5.normalize_coordinates() 8 sage: [[f.conjugate(m)==f for m in [m1,m2]] for f in [f7,f11,f5]] 9 [[True, True], [True, True], [True, True]] Benjamin Hutz for Dynamical Systems 39 / 48

40 Rational preperiodic points Post-critically finite maps Automorphism Groups Tetrahedral family: Family 1 sage: B.<t> = PolynomialRing(K) 2 sage: PB = P.change_ring(B) 3 sage: u,v = PB.gens() 4 sage: RB = PB.coordinate_ring() 5 sage: phib = R.hom([u,v],RB) 6 sage: f = DynamicalSystem([phiB(f7[0]*phi(p12a)) + t*phib(f11[0]*phi(p8)), phib(f7[1]*phi(p12a)) + t*phib(f11[1]*phi(p8))]) 7 sage: [f.conjugate(m)==f for m in [m1,m2]] 8 [True, True] 9 sage: f.resultant().factor() 10 C*tˆ12*(t + 2/3)ˆ18 Benjamin Hutz for Dynamical Systems 40 / 48

41 Products of projective space Iteration of subvarieties Example: P 2 P 1 1 sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2,1],QQ) 2 sage: F = DynamicalSystem([xˆ2*u,yˆ2*w,zˆ2*u,wˆ2,uˆ 2], domain=t) 3 sage: F(T([2,1,3,0,1])) 4 (4/9 : 0 : 1, 0 : 1) Benjamin Hutz for Dynamical Systems 41 / 48

42 Products of projective space Iteration of subvarieties WehlerK3 [CS96, Sil91, dfh15] 1 sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2,2],GF(7)) 2 sage: Q = x0ˆ2*y0ˆ2 + 3*x0*x1*y0ˆ2 + x1ˆ2*y0ˆ2 + 4* x0ˆ2*y0*y1 + 3*x0*x1*y0*y1-2*x2ˆ2*y0*y1 - x0ˆ2 *y1ˆ2 + 2*x1ˆ2*y1ˆ2 - x0*x2*y1ˆ2-4*x1*x2*y1ˆ2 + 5*x0*x2*y0*y2-4*x1*x2*y0*y2 + 7*x0ˆ2*y1*y2 + 4*x1ˆ2*y1*y2 + x0*x1*y2ˆ2 + 3*x2ˆ2*y2ˆ2 3 sage: L = x0*y0 + x1*y1 + x2*y2 4 sage: X = WehlerK3Surface([L,Q]) 5 sage: T = X([0,0,1,1,0,0]) 6 sage: X.orbit_phi(T,5) 7 [(0 : 0 : 1, 1 : 0 : 0), 8 (6 : 0 : 1, 0 : 1 : 0), 9 (4 : 4 : 1, 1 : 4 : 1), 10 (6 : 0 : 1, 1 : 4 : 1), 11 (0 : 0 : 1, 0 : 1 : 0), 12 (0 : 0 : 1, 1 : 0 : 0)] 13 sage: X.cardinality() Benjamin Hutz for Dynamical Systems 42 / 48

43 Products of projective space Iteration of subvarieties WehlerK3: Canonical Heights 1 sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(QQ,6) 2 sage: L = (-y0 - y1)*x0 + (-y0*x1 - y2*x2) 3 sage: Q = (-y2*y0 - y1ˆ2)*x0ˆ2 + ((-y0ˆ2 - y2*y0 + (-y2*y1 - y2ˆ2))*x1 +(-y0ˆ2 - y2*y1)*x2)*x0 + ( (-y0ˆ2 - y2*y0 - y2ˆ2)*x1ˆ2 + (-y2*y0 - y1ˆ2)* x2*x1 + (-y0ˆ2 + (-y1 - y2)*y0)*x2ˆ2) 4 sage: X = WehlerK3Surface([L,Q]) 5 sage: P = X([1,0,-1,1,-1,0]) #order 16 6 sage: X.canonical_height(P,5) Benjamin Hutz for Dynamical Systems 43 / 48

44 Products of projective space Iteration of subvarieties Computing the forward image Proposition Let f : P N P N be a morphism and X = V (g 1,..., g k ) be a subvariety. Let I K [x 0,..., x N, y 0,..., y N ] be the ideal I = (g 1 (x),..., g k (x), y 0 f 0 (x),..., y N f N (x)). Then the elimination ideal I N+1 = I K [y 0,..., y N ] is a homogeneous ideal and f (X) = V (I N+1 ). Benjamin Hutz for Dynamical Systems 44 / 48

45 Products of projective space Iteration of subvarieties Sanity Check: Iteration of points 1 sage: P.<x,y,z> = ProjectiveSpace(QQ,2) 2 sage: f = DynamicalSystem([xˆ2 -zˆ2,yˆ2-2*zˆ2,zˆ2]) 3 sage: X = P.subscheme([x, y-2*z]) 4 sage: [Z.defining_polynomials() for Z in f.orbit(x, 2)] 5 [(x, y - 2*z), (y - 2*z, x + z), (y - 2*z, x)] 6 sage: f.orbit(p([0,2,1]),2) 7 [(0 : 2 : 1), (-1 : 2 : 1), (0 : 2 : 1)] Benjamin Hutz for Dynamical Systems 45 / 48

46 Products of projective space Iteration of subvarieties Example: Periodic 1 sage: P.<x,y,z> = ProjectiveSpace(QQ,2) 2 sage: f = DynamicalSystem([xˆ2 + 6*x*z + 9*zˆ2, -x* z - 3/2*zˆ2, 4*x*y + 12*yˆ2-2*x*z - 3*zˆ2]) 3 sage: X = P.subscheme(x) 4 sage: [Z.defining_polynomials() for Z in f.orbit(x, 3)] 5 [(x,), (x + 6*y,), (x + 3*z,), (x,)] Benjamin Hutz for Dynamical Systems 46 / 48

47 Products of projective space Iteration of subvarieties Example: PCF 1 sage: P.<x,y,z> = ProjectiveSpace(QQ,2) 2 sage: f = DynamicalSystem([(x-2*y)ˆ2,(x-2*z)ˆ2,xˆ2] ) #fornaess 3 sage: X = P.subscheme(f.wronskian_ideal()) 4 sage: IC = X.irreducible_components() 5 sage: for X in IC: 6 sage: [Z.defining_polynomials() for Z in f. orbit(x,5)] 7 [(x,), (z,),(y - z,), (x - y,), (x - z,), (y - z,)] 8 [(x-2z,),(y,),(x - z,),(y - z,),(x - y,),(x - z,)] 9 [(x - 2y,),(x,),(z,),(y - z,), (x - y,), (x - z,)] Benjamin Hutz for Dynamical Systems 47 / 48

48 Products of projective space Iteration of subvarieties References [CS96] Gregory S. Call and Joseph H. Silverman, Computing the canonical height on K3 surfaces, Math. Comp. 65 (1996), [dfh15] Joao de Faria and Benjamim Hutz, Cycles on Wehler K3 surfaces, New Zealand Journal of Mathematics 45 (2015), [dfh17], Automorphism groups and invariant theory on PN, Journal of Algebra and its Applications forthcoming (2017). [FMV14] Xander Faber, Michelle Manes, and Bianca Viray, Computing conjugating sets and automorphism groups of rational functions, Journal of Algebra 423 (2014), [HT15] [Hut15] [Sil91] Benjamim Hutz and Adam Towsley, Thurston s theorem and misiurewicz points for polynomial maps, New York J. Math 21 (2015), Benjamin Hutz, Determination of all rational preperiodic points for morphisms of PN, Mathematics of Comptutaion 84 (2015), no. 291, Joseph H. Silverman, Rational points on K3 surfaces: a new cannonical height, Invent. Math. 105 (1991), Benjamin Hutz for Dynamical Systems 48 / 48