Bigraded Castelnuovo-Mumford regularity and Gröbner bases

TL;DR

The paper addresses bounding the bidegrees of generators in bigraded Gröbner bases for bihomogeneous ideals in $S=\mathbf{k}[x_0,\dots,x_n,y_0,\dots,y_m]$ by analyzing the bigraded Castelnuovo-Mumford regularity and introducing a partial regularity region $\xreg(I)$ defined through the vanishing of local cohomology with respect to the $x$-block. It proves a Bayer–Stillman–type criterion for $\xreg(I)$, shows that in generic coordinates $\xreg(I)=\xreg(\mathrm{bigin}(I))$ under the degree reverse lexicographic order with the $x$-block preceding the $y$-block, and uses this to bound and certify the presence of generators of $\mathrm{bigin}(I)$, while relating these bounds to $\reg(I)$ and Betti numbers via Chardin–Holanda. The main contributions are (i) explicit examples showing the limitations of directly extending single-graded reg-based bounds to the bihomogeneous setting, (ii) the introduction of $\xreg(I)$ as a sharp bounding tool for the $x$-degrees of generators of $\mathrm{bigin}(I)$, and (iii) a framework linking $\xreg(I)$, $\reg(I)$, and Betti numbers to better understand the complexity of computing bihomogeneous Gröbner bases. These results offer a more nuanced, partially sharp description of generator degrees and lay groundwork for multihomogeneous extensions and complexity estimates in Gröbner basis computations.

Abstract

We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal $I$ in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of $I$ with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for $I$. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of $I$.

Figures (8)

• Figure 1: The green dots represent the degrees of the generators of $I$ and the black dots represent the degrees of the generators of the bigeneric initial ideals. In this example $\xtor(I) = 8$ (in purple). In the region (in brown), which corresponds to a shift of $\xreg(I)$, we can certify that there are no minimal generators of $\bigin(I)$ of those bidegrees. Moreover, $\xreg(I)$ can be used to certify the presence of generators of $\bigin(I)$ of the bidegrees marked in blue.
• Figure 1: The green dots represent the degrees of the generators of $I$ and the black dots represent the degrees of the generators of the bigeneric initial ideals.
• Figure 1: The green dots $\bullet$ represent the bidegrees $(a,b)$ of the generators of the ideal $I$ in Example \ref{['examplemat2']}. The black dots $\bullet$ represent the bidegrees $(a,b)$ of minimal generators of $\bigin(I)$ and the white dots those bidegrees for which $\HF_{S/I}(a,b) = 0$. The region $\reg(I)$ is marked in red. In blue (resp. brown), an infinite column (resp. a row) which does not intersect $\reg(I)$.
• Figure 1: In olive, the region $\yreg(I)$. In blue, the region $\yreg(\bigin(I))$.
• Figure 1: In brown, the region $\xreg(I) + (1,0)$. In blue, columns and squares where there are generators of $\bigin(I)$.

Theorems & Definitions (77)

• Definition 1
• Example 2
• Definition 1
• Definition 2
• remark 1
• Definition 3
• Definition 4
• Lemma 5: Bigeneric initial ideal aramova2000bigeneric
• Lemma 6: aramova2000bigeneric
• remark 2