Suppose you have a homogeneous ideal I inside the algebra C[x\_1,...,x\_d]  of complex polynomials in d-variables. Can one find a basis for I, say {f\_1,...,f\_k}, such that every h in I can be written as 

h = a\_1\*f\_1 + ... + a\_k\*f\_k

where the coefficients appearing in each summand a\_i\*f\_i are not much bigger then the coefficients appearing in h?
More specifically, given that {f\_1,...,f\_k} is a Groebner basis for I, can one modify the standard division algorithm so that one gets 
h = a\_1\*f\_1 + ... + a\_k\*f\_k
with controlled terms?

I will share that I found this possible in some special cases, for example when d=2, and I am now interested in the general question.

Note: My question begins *after* a basis has been found, I am not concerned here with the terrible complexity of actually computing a Groebner basis.