ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 28 Oct 2020 20:12:20 +0100All rational periodic pointshttps://ask.sagemath.org/question/54066/all-rational-periodic-points/Hello, I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes that I used are following:
A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem_affine([2*z^3-3*z^2+1/2])
x=f.dynatomic_polynomial(2)
x.factor()
With this I can find its dynatomic polynomial and factorize it and find rational roots of this polynomial. So this roots corresponds to periodic point of the polynomial of given period. In particular dynatomic polynomial is the polynomial of the form $$\phi_{n,f}(x)=\prod_{d|n}(f^d(x)-x)^{\mu(n/d)}$$ n is period, f is your polynomial and $\mu$ is the mobius function.
But with this code I can find periods up to 8 because of memory limit. The other code that I used is
R.<x> = QQ[]
K.<i> = NumberField(xˆ2+1)
A.<z> = AffineSpace(K,1)
f = DynamicalSystem([zˆ2+i], domain=A)
f.orbit(A(0),4)
But in fact it doesn't fit my purposes.
I have codes that I can get limited information. For example checking up to a period is not advisable. If you know a little bit arithmetic dynamics, you can see what I mean. Silverman-Morton conjecture plays an important role here.
I am waiting for your answers. Thank you so much.nomaddWed, 28 Oct 2020 20:12:20 +0100https://ask.sagemath.org/question/54066/Generate all the monic polynomials up to degree n with coefficients in Zp(p prime number) field and find all the irriducible polynomialshttps://ask.sagemath.org/question/47178/generate-all-the-monic-polynomials-up-to-degree-n-with-coefficients-in-zpp-prime-number-field-and-find-all-the-irriducible-polynomials/My code.
def POLYNOMIAL_OBTAINED_BY_RECURSION(p,n):
R=Zmod(p)
Z.<x>=PolynomialRing(R)
All=[]
if n==1:
h=[]
for i in range(0,p):
pol=x+i
h.append(pol)
return(pol)
All.append(Set(h))
else :
h=[]
for i in range(0,p):
Pol=POLYNOMIAL_OBTAINED_BY_RECURSION(p,n-1)+i*(x^(n-1))+x^n
h.append(Pol)
return(Pol)
All.append(Set(h))
return(All)
def MYSIEVE(n,All)
for d in range(1,floor(n/2)+1):
for j in range(d+1,n+1):
for polinomio1 in All[j]:
for polinomio2 in All[d]:
if polinomio1%polinomio2==0:
k={polinomio1}
All[j].difference(k)
return(All)
def project(p,n):
if not(is_prime(p)):
print('p is not prime',p)
R=Zmod(p)
Z.<x>=PolynomialRing(R)
All=[]
All=POLYNOMIAL_OBTAINED_BY_RECURSION(p,n)
All=MYSIEVE(n,All)
return(All)AlessandroDeSantisTue, 16 Jul 2019 11:57:01 +0200https://ask.sagemath.org/question/47178/Algebra of functions f:Z_3 -> Rhttps://ask.sagemath.org/question/45402/algebra-of-functions-fz_3-r/ I want to create an implementation of an algebra of functions with domain {0,1,2} and range in R. The sum and the product is the usual pointwise sum and product. I have the idea of represent it as a 3 dimensional vectors with the usual sum of vectors, bot I need implement a new product.
The implementation of this algebra will be used for create a polynomial system of equations that I need to solve, so the implementation should be compatible with the procedure indicated in the entry titled *Find algebraic solutions to system of polynomial equations*
If someone have any idea please help me.
Thanks in advance.JulioNAQSat, 09 Feb 2019 22:01:49 +0100https://ask.sagemath.org/question/45402/