I came across the following problem. It seems reasonable at first sight, but still I’m not entirely sure whether it is correct or not. Maybe you have a fun time solving it or finding a counterexample.
Setting. We work in a multivariate polynomial ring $K[X] = K[X_1,\dotsc,X_N]$ in $N \in \mathbb{N}^+$ variables over a field $K$. Let an ideal $I = (f_1, \dotsc, f_m) = (g_1, \dotsc, g_n) \subseteq K[X]$ given by two different bases. Being two bases of the same ideal, we can obviously express each $g_i$ as a combination of the $f_j$ and vice versa. Suppose that in one direction we have $g_i = \sum_{j=1}^m s_{ij}f_j$ with coefficients $s_{ij} \in K[X]$ of degree at most $d = \max_{i,j} \deg(s_{ij})$.
Now I am interested in the other direction. We know that there exists a representation $f_j = \sum_{i=1}^n t_{ji}g_i$ with $t_{ji} \in K[X]$.
Question. Is there a bound $\beta = \beta(N,\max_j \deg(f_j),\max_i \deg(g_i),d) \in \mathbb{N}$ such that $t_{ji}$ can be chosen to be of degree at most $\beta$?
Loosely speaking: If we know how one ideal basis in a polynomial ring can be converted to another, do we have a bound for the necessary coefficient degrees of an inverse conversion?
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Edit: In the original question, $\beta$ was erroneously not allowed to depend on the maximal degrees of the bases.