Newton to poly function for enhancement #38393

[Nathabolin]

Fixes #38393

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[bhutz]

Some comments here

• deprecate the return_conjugation in is_newton

• looks like QQbar should work for is_newton

• Newton -> a Newton map

• look at some of the example spacing: = and around input ,

• you should use keyword=True in examples

• don't need to extra variable V. Can just say; if self.is_newton():

• This is wrong. Probably need is_newton to be true by default and note that if you set it to be false, you can get wrong answers.

P.<x,y>=ProjectiveSpace(QQbar,1)
f=DynamicalSystem([x^4,y^4])
M,h=f.Newton_to_poly(return_conjugation=True)
M,h

• This should probably work too. Take a look at the field of definition of the returned conjugation. Note that replacing QQbar by a numberfield fails with the same error.

A.<z>=AffineSpace(QQbar,1)
F = z^3 + z+ 1
f = DynamicalSystem_affine(z - F/(F.derivative(z)))
print(f)
g=f.homogenize(1)
m=matrix(QQ,2,2,[3,4,1,2])
g=g.conjugate(m)
M,h=g.Newton_to_poly(return_conjugation=True)
M,h
g.conjugate(M)

[bhutz]

A couple comments on this

• still need to return the conjugation until we're done deprecating
• note the failing doc tests that needs to be fixed due to the deprecation
• looking at the code I see two options for the field of definitions
  • keep M, self, and poly in their minimal fields of definition and return both embeddings to their compositum
  • move M, poly to the compositum and return the embedding that moves self to the compositum
    I think the second option is probably a simpler experience for the users, so is my preference.