Sage Basic Dynamics Functionality Computational Problems Other topics

Sage 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-1415294

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Sage Basic Dynamics Functionality Computational Problems Other topics

What is Sage? SageMath 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. http://www.sagemath.org/ Benjamin Hutz

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How can you use Sage

Online https://cloud.sagemath.com Locally Download from http://www.sagemath.org/ Prebuilt binaries Install from source code

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Dynamical Systems and Sage Sage-dynamics project 1

Started at ICERM in 2012

2

Sage-days 55 in 2013

3

NSF grant DMS-1415294, 2014-2017 (PI: Hutz)

4

Future Sage-days ?2017/2018?

Resources Project Page: http://wiki.sagemath.org/ dynamics/ArithmeticAndComplex Google group: sage-dynamics

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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

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Basic Examples: Iteration 1 2 3 4 5 6 7 8 9 10 11 12

sage: A. = AffineSpace(QQ,1) sage: H = End(A) sage: f = H([xˆ2-1]) sage: P=A(2) sage: f(P), f(f(P)), f(f(f(P))), f.nth_iterate(P,4) ((3), (8), (63), (3968)) sage: f.orbit(P,[0,3]) [(2), (3), (8), (63)] sage: f.nth_iterate_map(2) Scheme endomorphism of Affine Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x) to (xˆ4 - 2*xˆ2)

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Basic Examples: Varieties sage: A. = AffineSpace(QQ,2) sage: X = A.subscheme([x-1, yˆ2-4]); sage: X.dimension() 0 sage: X.rational_points() [(1, -2), (1, 2)] 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) 1 2 3 4 5 6 7

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Height definitions Given a point P = (P0 : · · · : PN ) The (absolute) global height is defined as h(P) =

1 log(max(|Pi |)) [K : Q]

The local height (Green’s function) at a place v is defined as λv (P) = log(max(|Pi |v )) The canonical height with respect to a morphism f is defined as h(f n (P)) ˆ h(P) = lim . n→∞ deg(f )n Benjamin Hutz

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Sage Basic Dynamics Functionality Computational Problems Other topics

Basic Examples: Heights

1 2 3 4 5 6 7 8 9 10

sage: P. = ProjectiveSpace(QQ,1) sage: H = End(P) sage: f = H([4*xˆ2-3*yˆ2,4*yˆ2]) sage: Q = P(3,5) sage: Q.green_function(f,2, prec=100, N=20) -0.69314585848661813056646800473 sage: Q.global_height() 1.60943791243410 sage: Q.canonical_height(f,error_bound=0.001) 2.3030927691516823627114122790

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Sage Basic Dynamics Functionality Computational Problems Other topics

Basic Examples: Local Heights

sage: K. = QuadraticField(3) sage: P. = ProjectiveSpace(K,1) sage: H = Hom(P,P) sage: f = H([17*xˆ2+1/7*yˆ2,17*w*x*y]) sage: f.green_function(P([2,1]), K.ideal(7), N=7) 6 0.48647753726382832627633818586 7 sage: f.green_function(P([w,1]), K.ideal(17), error_bound=0.001) 8 -0.70691993106090157426711999977 1 2 3 4 5

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Definitions: Dynatomic Polynomial Let f ∈ K [z] be a single variable polynomial. Definition The n-th dynatomic polynomial for f is defined as Y Φ∗n (f ) = (f d (z) − z)µ(n/d) . d|n

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

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Basic Examples: Dynatomic Polynomial 1 2 3 4 5 6

7 8 9 10

sage: R. = FunctionField(QQ) sage: P. = ProjectiveSpace(R,1) sage: H = End(P) sage: f = H([xˆ2+c*yˆ2,yˆ2]) sage: f.dynatomic_polynomial(3) 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 sage: f.dynatomic_polynomial(3).subs({x:0,y:1}) cˆ3 + 2*cˆ2 + c + 1 sage: f.nth_iterate(P(0),3) (cˆ4 + 2*cˆ3 + cˆ2 + c : 1)

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Sage Basic Dynamics Functionality Computational Problems Other topics

Basic Examples: Number Fields

1 2 3 4 5 6 7 8 9 10

sage: A. = AffineSpace(QQ,1) sage: H = End(A) sage: f = H([zˆ2+1]) sage: R = A.coordinate_ring() sage: K. = NumberField(f.dynatomic_polynomial(2) .polynomial(R.gen(0))) sage: A. = AffineSpace(K,1) sage: H = End(A) sage: f = H([zˆ2+1]) sage: f.orbit(A(w),3) [(w), (-w - 1), (w), (-w - 1)]

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Sage Basic Dynamics Functionality Computational Problems Other topics

Basic Examples: All Periodic Points

sage: P. = ProjectiveSpace(QQbar,1) sage: H = End(P) sage: f = H([xˆ2+yˆ2,yˆ2]) sage: f.periodic_points(2) 5 [(-0.500000000000000? - 1.322875655532296?*I : 1), 6 (-0.500000000000000? + 1.322875655532296?*I : 1)] 1 2 3 4

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Sage Basic Dynamics Functionality Computational Problems Other topics

Basic Examples: Critical Points

sage: P. = ProjectiveSpace(QQbar,1) sage: H = End(P) sage: f = H([xˆ2+yˆ2,yˆ2]) sage: f.critical_points() 5 [(0 : 1), (1 : 0)] 6 sage: f.critical_height(error_bound=0.001) 7 0.20367726136974000143661899538 1 2 3 4

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Sage Basic Dynamics Functionality Computational Problems Other topics

Basic Examples: Finite Fields

sage: P. = ProjectiveSpace(GF(13),1) sage: H = End(P) sage: f = H([xˆ2+yˆ2,yˆ2]) sage: f.cyclegraph() 5 Looped digraph on 14 vertices 6 sage: P([2,1]).orbit_structure(f) 7 [0, 4] 1 2 3 4

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1

Rational preperiodic points 1

2 2

2

Determine if a morphism f : P1 → P1 is post-critically finite Construct a pcf map with specified portrait

Automorphism groups 1

4

Determine all of the rational preperiodic points for a morphism f : PN → PN . Determine if a given point is preperiodic for f .

Post-critically finite maps 1

3

Rational preperiodic points Post-critically finite maps Automorphism Groups

Construct a family of maps each of whose automorphism group contains the given finite group.

Other topics 1 2

Benjamin Hutz

Products of projective spaces and Wehler K3 surfaces Iteration of subvarieties

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Definitions Theorem (Northcott) The set of rational preperiodic points for a morphism f : PN → PN is a set of bounded height. Conjecture (Morton-Silverman) Given a morphism f : PN → PN 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).

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A precise description of n Theorem (Morton-Silverman 1994, Zieve) Let f : P1 → P1 defined over Q with deg(f ) ≥ 2. Assume that f has good reduction at p with a rational periodic point P. Define n = minimal period of P. m = minimal period of P modulo p. r = the multiplicative order of (f m )0 (P) mod p Then n=m

or

n = mrpe

for some explicitly bounded integer e ≥ 0.

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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 2

2

3

Intersect the lists of possible periods for all primes in PrimeSet. For each n in PossiblePeriods 1

4 5

Find all of the periodic cycles modulo p Compute m, mrpe for each cycle.

Find all rational solutions to f n (x) = x.

Let PrePeriodicPoints = PeriodicPoints. 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

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Examples sage: P. = ProjectiveSpace(QQ,1) sage: H = End(P) sage: f = H([xˆ2-29/16*yˆ2,yˆ2]) 4 sage: f.rational_preperiodic_graph() 5 Looped digraph on 9 vertices 1 2 3

1 2 3 4 5 6 7

sage: R. = PolynomialRing(QQ) sage: K. = NumberField(tˆ3 + 16*tˆ2 - 10496*t + 94208) sage: PS. = ProjectiveSpace(K,1) sage: H = End(PS) sage: f = H([xˆ2-29/16*yˆ2,yˆ2]) #Hutz sage: len(f.rational_preperiodic_points()) #6s 21

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Examples

3 4 5 6

sage: sage: sage: sage: sage: 13

K. = QuadraticField(33) PS. = ProjectiveSpace(K,1) H = End(PS) f = H([xˆ2-71/48*yˆ2,yˆ2]) #Stoll len(f.rational_preperiodic_points()) #1s

1 2 3 4 5

sage: sage: sage: sage: 44

P. = ProjectiveSpace(QQ,2) H = End(P) f = H([xˆ2 - 21/16*zˆ2,yˆ2-2*zˆ2,zˆ2]) len(f.rational_preperiodic_points()) #1.5s

1 2

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Is a point preperiodic? Let f : PN → PN be a morphism of degree d. Theorem ˆ A point P is preperiodic for f if and only if h(P) = 0. Theorem There exists a constant C independent of a point P such that ˆ h(P) − h(P) ≤ C(f ). Theorem For all points P ˆ (P)) = d h(P). ˆ h(f Benjamin Hutz

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Algorithm: is preperiodic()

1 2

ˆ If h(P) > C, then not preperiodic. Compute forward images 1 2

Benjamin Hutz

If we encounter a cycle, return preperiodic If the height becomes > C, return not preperiodic.

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Example sage: sage: sage: 4 sage: 5 sage: 6 (1,3)

P. = ProjectiveSpace(QQ,2) H = End(P) f = H([xˆ2-29/16*zˆ2,yˆ2 - 21/16*zˆ2,zˆ2]) Q = P(1,-3,4) Q.is_preperiodic(f,return_period=True)

sage: sage: sage: sage: sage: sage: False

K. = QuadraticField(-2) P. = ProjectiveSpace(K,2) H = End(P) f = H([xˆ2-29/16*zˆ2,yˆ2 - 21/16*zˆ2,zˆ2]) Q = P(w,-3,4) Q.is_preperiodic(f)

1 2 3

1 2 3 4 5 6 7

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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 X ˆ h(c). c∈crit Benjamin Hutz

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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.

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Example

1 2 3 4 5 6 7 8 9

sage: P. = ProjectiveSpace(QQ,1) sage: H = End(P) sage: f = H([xˆ2 + 12*yˆ2, 7*x*y]) sage: f.critical_points(R=QQbar) [(-3.464101615137755? : 1), (3.464101615137755? : 1 )] sage: f.is_postcritically_finite() False sage: f.critical_height(error_bound=0.001) 2.8717614996729500069637701410

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` maps Example: Lattes

1 2 3 4 5 6 7 8 9 10

sage: P. = ProjectiveSpace(QQ,1) sage: E = EllipticCurve([0,0,0,0,2]);E Elliptic Curve defined by yˆ2 = xˆ3 + 2 over Rational Field sage: f = P.Lattes_map(E,2) sage: f.is_postcritically_finite() True sage: f.critical_point_portrait() Looped digraph on 10 vertices sage: f.critical_height(error_bound=0.001) 8.2900752025070323779826707582e-17

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Misiurewicz Point sage: sage: sage: 4 sage: 5 sage: 6 cˆ2+1 1 2 3

1 2 3 4 5 6 7

sage: sage: sage: sage: sage: sage: [(0),

Benjamin Hutz

R. = PolynomialRing(QQ) A. = AffineSpace(R,1) H = End(A) f = H([zˆ2+c]) f.dynatomic_polynomial([2,2]).subs({z:0})

R. = QQ[] K. = NumberField(xˆ2+1) A. = AffineSpace(K,1) H = End(A) f = H([zˆ2+i]) f.orbit(A(0),4) (i), (i - 1), (-i), (i - 1)] Sage for Dynamical Systems

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Gleason Polynomial

1 2 3 4 5 6 7 8 9 10

sage: R. = PolynomialRing(QQ) sage: A. = AffineSpace(R,1) sage: H = End(A) sage: f = H([zˆ2+c]) sage: K. = NumberField(f.dynatomic_polynomial(3) .subs({z:0})) sage: A. = AffineSpace(K,1) sage: H = End(A) sage: f = H([zˆ2+w]) sage: f.orbit(A(0),3) [(0), (w), (wˆ2 + w), (0)]

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Rational preperiodic points Post-critically finite maps Automorphism Groups

2-parameters [HT15]: generate ideal

sage: R. = PolynomialRing(QQ,2, order='lex') sage: P. = ProjectiveSpace(R,1) sage: H = End(P) sage: f = H([xˆ3 -3*aˆ2*x*yˆ2 +(2*aˆ3+v)*yˆ3,yˆ3]) sage: dyn1 = f.dynatomic_polynomial(2) 6 sage: dyn2 = f.dynatomic_polynomial(1) 7 sage: I = R.ideal([dyn1.subs({x:a,y:1}), dyn2.subs( {x:-a,y:1})]) 1 2 3 4 5

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2-parameters: get solutions

1 2 3 4 5 6 7 8 9

sage: sage: sage: sage: sage: sage: sage: sage: sage:

Benjamin Hutz

G = I.groebner_basis() S. = QQ[] phi = R.hom([0,z],S) p = phi(G[-1]).factor() K. = NumberField(p[0][0]) S. = K[] phi2 = R.hom([z,V],S) q = phi2(G[-2]).factor() L. = K.extension(q[0][0])

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2-parameters: check orbits

sage: P.=ProjectiveSpace(L,1) sage: H=End(P) sage: f=H([xˆ3 -3*Aˆ2*x*yˆ2 +(2*Aˆ3+V)*yˆ3,yˆ3]) sage: f.orbit(P(A),2) [(2/3*Vˆ5 + Vˆ3 + 5/6*V : 1), (V : 1), (2/3*Vˆ5 + V ˆ3 + 5/6*V : 1)] 6 sage: f.orbit(P(-A),2) 7 [(-2/3*Vˆ5 - Vˆ3 - 5/6*V : 1), 8 (-2/3*Vˆ5 - Vˆ3 - 5/6*V : 1)] 1 2 3 4 5

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Definitions

Definition Let f : PN → PN . Then an element α ∈ PGLN+1 acts on f via conjugation f α = α ◦ f ◦ α−1 Definition The automorphism group of f is the group Aut(f ) = {α ∈ PGLN+1 : f α = f }.

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Determining Automorphism Groups [FMV14]

sage: R. = ProjectiveSpace(QQ,1) sage: H = End(R) sage: f = H([xˆ2-2*x*y-2*yˆ2,-2*xˆ2-2*x*y+yˆ2]) sage: f.automorphism_group(return_functions=True) [x, 2/(2*x), -x - 1, -2*x/(2*x + 2), (-x - 1)/x, -1 /(x + 1)] 6 sage: f.conjugate(matrix([[-1,-1],[0,1]])) == f 7 True 1 2 3 4 5

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Constructing Automorphisms [dFH15a] Definition Given a finite matrix group G, we say that a polynomial p is an invariant of G if p◦g =p

for all g ∈ G.

Theorem (Klein) If F is an invariant for a finite group G, then f = (−Fy : Fx ) : P1 → P1 has G ⊂ Aut(f ). Benjamin Hutz

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Sage Basic Dynamics Functionality Computational Problems Other topics

Rational preperiodic points Post-critically finite maps Automorphism Groups

Tetrahedral family: Invariants 1 2 3 4 5 6 7 8 9 10 11

sage: R. = QQ[] sage: K. = CyclotomicField(4) sage: m1 = matrix(K,2,[(-1+i)/2,(-1+i)/2, (1+i)/2,( -1-i)/2]) sage: m2 = matrix(K,2, [0,i, -i,0]) sage: tetra = MatrixGroup(m1,m2) sage: [p8,p12a,p12b] = tetra.invariant_generators() sage: S = p8.parent() sage: x1,x2 = S.gens() sage: F7 = [-p8.derivative(x2),p8.derivative(x1)] sage: F11 = [-p12a.derivative(x2),p12a.derivative( x1)] sage: F5 = [-p12b.derivative(x2),p12b.derivative(x1 )]

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Sage Basic Dynamics Functionality Computational Problems Other topics

Rational preperiodic points Post-critically finite maps Automorphism Groups

Tetrahedral family: Morphisms

1 2 3 4 5 6 7 8 9 10

sage: P. = ProjectiveSpace(K,1) sage: H = End(P) sage: R = P.coordinate_ring() sage: phi = S.hom([x,y],R) sage: f7 = H([phi(t) for t in F7]) sage: f11 = H([phi(t) for t in F11]) sage: f5 = H([phi(t) for t in F5]) sage: f5.normalize_coordinates() sage: [[f.conjugate(m)==f for m in [m1,m2]] for f in [f7,f11,f5]] [[True, True], [True, True], [True, True]]

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Rational preperiodic points Post-critically finite maps Automorphism Groups

Tetrahedral family: Family 1 2 3 4 5 6 7

8 9 10 11

sage: B. = PolynomialRing(K) sage: PB = P.change_ring(B) sage: u,v = PB.gens() sage: RB = PB.coordinate_ring() sage: phiB = R.hom([u,v],RB) sage: HB = End(PB) sage: f = HB([phiB(f7[0]*phi(p12a)) + t*phiB(f11[0] *phi(p8)), phiB(f7[1]*phi(p12a)) + t*phiB(f11[1 ]*phi(p8))]) sage: [f.conjugate(m)==f for m in [m1,m2]] [True, True] sage: f.resultant().factor() C*tˆ12*(t + 2/3)ˆ18

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Sage Basic Dynamics Functionality Computational Problems Other topics

Products of projective space Iteration of subvarieties

Example: P2 × P1

1 2 3 4 5

sage: T. = ProductProjectiveSpaces([2,1] ,QQ) sage: H = End(T) sage: F = H([xˆ2*u,yˆ2*w,zˆ2*u,wˆ2,uˆ2]) sage: F(T([2,1,3,0,1])) (4/9 : 0 : 1 , 0 : 1)

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Sage Basic Dynamics Functionality Computational Problems Other topics

Products of projective space Iteration of subvarieties

WehlerK3 [CS96, Sil91, dFH15b] 1 2

3 4 5 6 7 8 9 10 11 12 13 14

sage: PP. = ProductProjectiveSpaces([2,2],GF(7)) 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 sage: L = x0*y0 + x1*y1 + x2*y2 sage: X = WehlerK3Surface([L,Q]) sage: T = X([0,0,1,1,0,0]) sage: X.orbit_phi(T,5) [(0 : 0 : 1 , 1 : 0 : 0), (6 : 0 : 1 , 0 : 1 : 0), (4 : 4 : 1 , 1 : 4 : 1), (6 : 0 : 1 , 1 : 4 : 1), (0 : 0 : 1 , 0 : 1 : 0), (0 : 0 : 1 , 1 : 0 : 0)] sage: X.cardinality() 55

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Products of projective space Iteration of subvarieties

WehlerK3: Canonical Heights

1 2 3

4 5 6 7

sage: R. = PolynomialRing(QQ,6) sage: L = (-y0 - y1)*x0 + (-y0*x1 - y2*x2) 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) sage: X = WehlerK3Surface([L,Q]) sage: P = X([1,0,-1,1,-1,0]) #order 16 sage: X.canonical_height(P,5) 0.00000000000000000000000000000

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Sage Basic Dynamics Functionality Computational Problems Other topics

Products of projective space Iteration of subvarieties

Computing the forward image

Proposition Let f : PN → PN be a morphism and X = V (g1 , . . . , gk ) be a subvariety. Let I ⊂ K [x0 , . . . , xN , y0 , . . . , yN ] be the ideal I = (g1 (x), . . . , gk (x), y0 − f0 (x), . . . , yN − fN (x)). Then the elimination ideal IN+1 = I ∩ K [y0 , . . . , yN ] is a homogeneous ideal and f (X ) = V (IN+1 ).

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Sage Basic Dynamics Functionality Computational Problems Other topics

Products of projective space Iteration of subvarieties

Sanity Check: Iteration of points sage: P. = ProjectiveSpace(QQ,2) sage: H = End(P) sage: f = H([xˆ2 -zˆ2,yˆ2-2*zˆ2,zˆ2]) sage: X = P.subscheme([x, y-2*z]) sage: [Z.defining_polynomials() for Z in f.orbit(X, 2)] 6 [(x, y - 2*z), (y - 2*z, x + z), (y - 2*z, x)] 7 sage: f.orbit(P([0,2,1]),2) 8 [(0 : 2 : 1), (-1 : 2 : 1), (0 : 2 : 1)] 1 2 3 4 5

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Sage Basic Dynamics Functionality Computational Problems Other topics

Products of projective space Iteration of subvarieties

Example: Periodic sage: P. = ProjectiveSpace(QQ,2) sage: H = End(P) sage: f = H([xˆ2 + 6*x*z + 9*zˆ2, -x*z - 3/2*zˆ2, 4 *x*y + 12*yˆ2 - 2*x*z - 3*zˆ2]) 4 sage: X = P.subscheme(x) 5 sage: [Z.defining_polynomials() for Z in f.orbit(X, 3)] 6 [(x,), (x + 6*y,), (x + 3*z,), (x,)] 1 2 3

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Sage Basic Dynamics Functionality Computational Problems Other topics

Products of projective space Iteration of subvarieties

Example: PCF 1 2 3 4 5 6 7 8 9 10

sage: P. = ProjectiveSpace(QQ,2) sage: H = End(P) sage: f = H([(x-2*y)ˆ2,(x-2*z)ˆ2,xˆ2]) #fornaess sage: X = P.subscheme(f.wronskian_ideal()) sage: IC = X.irreducible_components() sage: for X in IC: sage: [Z.defining_polynomials() for Z in f. orbit(X,5)] [(x,), (z,),(y - z,), (x - y,), (x - z,), (y - z,)] [(x-2z,),(y,),(x - z,),(y - z,),(x - y,),(x - z,)] [(x - 2y,),(x,),(z,),(y - z,), (x - y,), (x - z,)]

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Sage Basic Dynamics Functionality Computational Problems Other topics

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), 259–290.

[dFH15a] Joao de Faria and Benjamim Hutz, Automorphism groups and invariant theory on PN, arXiv:1509.06670. [dFH15b]

, Cycles on Wehler K3 surfaces, New Zealand Journal of Mathematics 45 (2015), 19–31.

[FMV14] Xander Faber, Michelle Manes, and Bianca Viray, Computing conjugating sets and automorphism groups of rational functions, Journal of Algebra 423 (2014), 1161–1190. [HT15]

Benjamim Hutz and Adam Towsley, Thurston’s theorem and misiurewicz points for polynomial maps, New York J. Math 21 (2015), 197–319.

[Hut15]

Benjamin Hutz, Determination of all rational preperiodic points for morphisms of PN, Mathematics of Comptutaion 84 (2015), no. 291, 289–308.

[Sil91]

Joseph H. Silverman, Rational points on K3 surfaces: a new cannonical height, Invent. Math. 105 (1991), 347–373.

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