### Can we avoid to fall back to very slow toy implementation in the computation of Groebner basis under a finite field of large characteristic?

I made a computation involving the computation of a Groebner basis under a finite field of large characteristic $p$.

If $p<2^{31}$ then everything is ok:

```
sage: %time function(previous_prime(2^31))
CPU times: user
```~~875 ~~15 ms, sys: 16 ms, total: 31 ~~ms, total: 906 ~~ms
Wall time: ~~1.14 s
[1]
~~28.3 ms
[v0 + 357913941, v1 - 357913941]

But if $p>2^{31}$ then the computation became very slow:

```
sage: %time function(next_prime(2^31))
verbose 0 (3837: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy
```~~implementation.
# 6 times above message
~~implementation.
CPU times: user ~~36min 28s, ~~125 ms, sys: ~~9.23 s, ~~219 ms, total: ~~36min 38s
~~344 ms
Wall time: ~~37min 27s
[1]
~~3.09 s
[v0 + 357913943, v1 + 1789569716]

I guess that it has something to do with 32-bits computer integers. The point is that my computer is 64-bits.

**Question**: Is there a way to avoid this slow-down (at least for $p<2^{63}$)?

Below is the code for `function`

(it is a toy example):

```
def function(p):
F=GF(p)
R.<v0, v1>=PolynomialRing(F,2)
Eq=[3*v0 + 5*v1 + 1/F(3),
9*v0^2 + 15*v1^2 - 2/F(3),
3*v0^3 + 5*v1^3 - v0^2 + 1/F(27)]
Id=Ideal(Eq)
G=Id.groebner_basis()
return G
```