Each polynomial can be seen as a dictionary whose items
encode the monomials.

The keys are tuple of degrees and the values are the
corresponding coefficients.

We can filter that dictionary, keeping only
monomials of bounded total degree.

Then we can build the polynomial corresponding
to that filtered dictionary.

Here is a function to do that:

```
def truncate(P, N):
r"""
Return this polynomial truncated to total degree N.
"""
trunc = {dd: c for dd, c in P.dict().items() if sum(dd) <= N}
return P.parent()(trunc)
```

Usage:

```
sage: R.<a, b, c> = PolynomialRing(QQ)
sage: PP = [R.random_element(5) for _ in range(3)]
sage: PP
[a*b^4 + 1/2*b^4*c + 2/3*b^3*c - 4/9*b^2*c - b^2,
5*b^5 - 3/2*a^3*c^2 + 3/5*b^2*c - 1,
-1/6*b^3*c^2 - 3*a*c^3 + b^2 - 3*b*c]
sage: [truncate(P, 3) for P in PP]
[-4/9*b^2*c - b^2, 3/5*b^2*c - 1, b^2 - 3*b*c]
```

The multivariate polynomials method `truncate`

only allows
to truncate with respect to the degree in one of the variables.
It could be improved to perform truncation by total degree.

See https://ask.sagemath.org/question/614...

Thank you Max. What if I want to truncate each polynomial to different precision? Is it possible in sage?