Complexity of finding algebraic dependency of polynomials over the rationals or in a finite field?

Question

Let $f_1,\ldots f_m \in K[x_1,\ldots,x_n]$ where $K$ is $\mathbb{Q}$ or a finite field.

Q1 What is the complexity of finding all algebraic dependencies between $f_i$?

Q2 What is the complexity of finding all algebraic dependencies between $f_i$ of degree up to $d$?

Sage can compute this and I believe it computes first Groebner basis, which is slow.

Today I am still waiting for dependency of only six low degree polynomials.

Couldn't find the answer with web searching.