Bug in dynatomic_polynomial

[David Krumm]

For a degree-2 endomorphism of the projective line, the second dynatomic polynomial should have degree 2. In the following example, Sage returns 0 as the second dynatomic polynomial. 

K.<a> = Frac(QQ['a'])
P.<x,y> = ProjectiveSpace(K, 1)
f = x^2 + (a + 1/a - 3)*x*y - (2*a + 1/a - 3)*y^2
phi = DynamicalSystem_projective([f,x^2])

phi.dynatomic_polynomial(2)

Moreover, Sage crashes when computing the third dynatomic polynomial; I get the message "Unhandled SIGSEGV: A segmentation fault occurred". Any idea what might be going on here?

-- David

[Ben]

Seems to be something with the quotient and remainder failing in the
fraction field of a polynomial ring:

phi2=phi.nth_iterate_map(2)
D2=x*phi2[1] - y*phi2[0]
D=x*phi[1] - y*phi[0]
print(D.divides(D2))
D2.quo_rem(D)


If you normalize to a polynomal ring it is working just fine:

K.<a> = QQ[]

P.<x,y> = ProjectiveSpace(K, 1)

f = a*x^2 + (a^2 + 1 - 3*a)*x*y - (2*a^2 + 1 - 3*a)*y^2
phi = DynamicalSystem_projective([f,a*x^2])
phi.dynatomic_polynomial(2)

I haven't dug down to see the underlying issue yet.

Ben

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[Ben]

There is an issue now tracking the underlying bug here:

https://github.com/sagemath/sage/issues/35886

On 6/23/2023 9:21 PM, David Krumm wrote: