Back in 2009, there was discussion on the Sage developers mailing list:

https://groups.google.com/forum/#!topic ... MPWGadoSpAabout the fact that in a multivariate polynomial ring of very small characteristic, raising a polynomial to a large power was being done suboptimally. Namely, one should really account for the fact that in characteristic p, raising to the p-th power acts on individual monomials; so to compute f^n, one ought to write n = a_0 + a_1 p + a_2 p^2 + ... and compute f^n as f^a_0 * (f^p)^(a_1) * ...

I just repeated the timing test from that thread, and as of 2016 it is still possible to do much better than what is implemented by explicitly splitting the exponent into base-p digits. It appears that the underlying representation of these polynomials is in Singular, so maybe the discussion should be continued here.

To make a particularly egregious example:

int p = 7;

ring rp = p, (x,y,z), dp;

poly f = (x^3 + y*z + z + y*y + x*z)^4;

poly g= f^(p^2);

should return instantaneously; instead, it takes about 10 seconds on my laptop.

Kiran