Dimension-dependent bounds for Gröbner bases of polynomial ideals
Let \(\mathbb{K}\) be an infinite field, \(\mathbb{K}[x_{1},\dots, x_{n}]\) an algebra of polynomials in \(n\) indeterminates over \(\mathbb{K}\), and \(I\) a proper ideal of \(\mathbb{K}[x_{1},\dots, x_{n}]\) of dimension \(r\) generated by a set of polynomials \(F = \{f_{1},\dots, f_{s}\}\) of degrees \(d_{1}\geq\dots\geq d_{s}\). The authors obtain the following bounds for the degree \(\deg(G)\) of the reduced Gröbner basis \(G\) of \(I\): 1) If the polynomials \(f_{1},\dots, f_{s}\) are homogeneous, then for any monomial ordering, \(\deg(G)\leq 2\left[\frac{1}{2}(d_{1}\dots d_{n-r} + d_{1})\right]^{2^{r-1}}\) (Theorem 33). 2) If \(f_{1},\dots, f_{s}\) are arbitrary polynomials, then for any monomial ordering, \[ \deg(G)\leq 2\left[{\frac{1}{2}}(d_{1}\dots d_{n-r})^{2(n-r)} + d_{1})\right]^{2^{r}} \text{ (Theorem 36).} \] Considering the problem of the lower degree bound, the authors prove that there are a monomial ordering and a family of ideals \(I_{r,n}\subseteq \mathbb{K}[x_{1},\dots, x_{n}]\) of dimension at most \(r\) for \(r, n\in \mathbb{N}\) with \(r\leq n\), which are generated by \(O(n)\) polynomials \(F_{r,n}\) of degrees bounded by \(d\) such that the degrees of the corresponding Gröbner bases \(G_{r,n}\) satisfy the condition \(\deg(G_{r,n})\geq d^{(n-r)2^{(1/2 - \epsilon)r}}\) for any \(\epsilon > 0\) and sufficiently large \(d, r\in \mathbb{N}\) (Theorem 38).
scientific article; zbMATH DE number 6124215
scientific article