Suppose you have a homogeneous ideal $I$ inside the algebra $\mathbb{C}[x\_1,...,x\_d]$ 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?


**Added 13.11.09 -** By *controlled* I mean that the **coefficients** of the terms $a\_i f\_i$ are bounded in a non-exponential manner by the coefficients of $h$. There is no problem with **degree** of the $a\_i$'s.


I will share that I found this possible in some special cases, for example when $d=2$ or when $I$ is generated by monomials, 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.


**Another note (added 12.11.09): The answers and links that I am getting suggest that this problem has not been considered before. So I re-eamphasize my note from above: assume that a Groebner basis, even a universal Groebner basis, has already been found for the ideal. What can be said about the stability of certain variants of the division algorithm now?** 



Is there a stable algorithm for polynomial division (in several variables)?
In Bayer and Mumford's [What Can Be Computed in Algebraic Geometry?](http://arxiv.org/pdf/alg-geom/9304003v1) section 3, you can find a 17 years old survey of known results. The bottom line is that without controlling the Castelnuovo Mumford regularity of the variety in question, there is very little you can do (you follow Buchberger's algorithm step by step and get some doubly exponential bound on the coefficients). The situation when imposing geometric conditions is a mater of current research.


