Computing Gröbner basis elements of some constant degree

Question

I'm wondering if there is any way or any special set of ideals such that there is an efficient way to compute elements of degree at most $d$ in a Gröbner basis for that ideal.

If you have any paper or hint I appreciate it.

Answer 1

For fixed $d$, you can do it in polynomial time. More precisely, Proposition 24.1 of Algorithmes efficaces en calcul formel by Bostan, Chyzak, Giusti, Lebreton, Lecerf, Salvy, Schost available here says

Given homogeneous polynomials $f_1,\dots,f_s$ in $k[X_0,\dots,X_n]$ and an integer $d$, there is an algorithm that computes the elements of degree at most $d$ of a Groebner basis of $I = (f_1,\dots,f_s)$ in time $$O\left(sd\binom{n+d}{d}^\omega\right)$$ where $2\le \omega \le 3$ is a matrix multiplication exponent.