# Subresultant algorithm taking a lot of time for higher degree univariate polynomials with coefficients from fraction fields

I have to compute the gcd of univariate polynomials over the fraction field of $\mathbb{Z}[x,y]$. I wanted to use the subresultant algorithm already implemented for UFDs. I copied the same function to fraction_field.py. The subresultant algorithm calls the psuedo division algorithm which has the following step :
R = d*R - c*B.shift(diffdeg) - this hangs when we consider random polynomials of degree >6 in $Frac(\mathbb{Z}[x,y])[z]$.

(Note: In the current version of sage it uses the regular Euclidean algorithm implemented in rings.py for computing gcd in this case. It is much slower than the subresultant algorithm (hangs for degrees >4) which is why I thought the subresultant algorithm will improve things.)

Sample input: sage: A.<x,y>=ZZ[]

sage: B= Frac(A)

sage: C.<z> = B[]

sage: p = C.random_element(6)

sage: q = C.random_element(6)

sage: gcd(p,q)

The following function is what I copied into fraction_field.py from unique_factorisation_domain.py.

def _gcd_univariate_polynomial(self, f, g):

```
if f.degree() < g.degree():
A,B = g, f
else:
A,B = f, g
if B.is_zero():
return A
a = b = self.zero()
for c in A.coefficients():
a = a.gcd(c)
if a.is_one():
break
for c in B.coefficients():
b = b.gcd(c)
if b.is_one():
break
d = a.gcd(b)
#d=1
A = A // a
B = B // b
g = h = 1
delta = A.degree()-B.degree()
_,R = A.pseudo_quo_rem(B)
while R.degree() > 0:
A = B
B = R // (g*h**delta)
g = A.leading_coefficient()
h = h*g**delta // h**delta
delta = A.degree() - B.degree()
_, R = A.pseudo_quo_rem(B)
# print("i am here")
if R.is_zero():
b = self.zero()
for c in B.coefficients():
b = b.gcd(c)
if b.is_one():
break
return d*B // b
return d
```

This calls the following pseudo quo remainder function in polynomial_element.pyx. It is in this function I was able to see that it hangs at R = d*R - c*B.shift(diffdeg).

def pseudo_quo_rem(self,other):

```
if other.is_zero():
raise ZeroDivisionError("Pseudo-division by zero is not possible")
# if other is a constant, then R = 0 and Q = self * other^(deg(self))
if other in self.parent().base_ring():
return (self * other**(self.degree()), self._parent.zero())
R = self
B = other
Q = self._parent.zero()
e = self.degree() - other.degree() + 1
d = B.leading_coefficient()
while not R.degree() < B.degree():
c = R.leading_coefficient()
diffdeg = R.degree() - B.degree()
Q = d*Q + self.parent()(c).shift(diffdeg)
R = d*R - c*B.shift(diffdeg)
e -= 1
q = d**e
return (q*Q,q*R)
```

It is really hard to reconstruct the error. (Without also copying the obvious same function to fraction_field, and guessing the code that builds the fraction fields, and knowing the meaning of d,R, cB, shift, diffdeg after an fgrep of potential modules, and searching / finding an explicit example that reproduces the same problem.) It would be simpler for a potential helper if the question already isolates a minimal example with simple polynomials also leading to the same error, and the code would be welcome...

yes I understand that. I will edit the question with more details.

I am the author of these lines. I don't have time to have a careful look right now, but I'll do that next week.

thank you! that will be great!