Dimension-Dependent Upper Bounds for Gröbner Bases

Gröbner bases, introduced by Bruno Buchberger in his Ph.D. thesis (see e.g. [6, 7]), have become a powerful tool for constructive problems in polynomial ideal theory and related domains. For practical applications, in particular, the implementation in computer algebra systems, it is important to establish upper bounds for the complexity of determining a Gröbner basis for a given homogeneous polynomial ideal. Using Lazard’s algorithm [23], a good measure to estimate such a bound, is an upper bound for the degree of the intermediate polynomials during the Gröbner basis computation. If the input ideal is not homogeneous, the maximal degree of the output Gröbner basis is not sufficient for this estimation. On the other hand, Möller and Mora [30] showed that to discuss degree bounds for Gröbner bases, one can restrict to homogeneous ideals. Thus upper bounds for the degrees of the elements of Gröbner bases of homogeneous ideals, allow us to estimate the complexity of computing Gröbner bases in general.

Let us review some of the existing results in this direction. Let $\mathcal{P}$ be the polynomial ring $\mathbbm{k}[x_{1},\ldots,x_{n}]$ where $\mathbbm{k}$ is of characteristic zero and $\mathcal{I}\subset\mathcal{P}$ be an ideal generated by homogeneous polynomials of degree at most $d$ with $\dim(\mathcal{I})=D$. The first doubly exponential upper bounds were proven by Bayer, Möller, Mora and Giusti, see [31, Chapter 38] for a comprehensive review of this topic. Based on results due to Bayer [2] and Galligo [14, 15], Möller and Mora [30] provided the upper bound $(2d)^{(2n+2)^{n+1}}$ for any Gröbner basis of $\mathcal{I}$. They also proved that this doubly exponential behavior cannot be improved. Simultaneously, Giusti [16] showed the upper bound $(2d)^{2^{n-1}}$ for the degree of the reduced Gröbner basis (w.r.t. the degree reverse lexicographic order) of $\mathcal{I}$ when the ideal is in generic position. Then, using a self-contained and constructive combinatorial argument, Dubé [10] proved the so far sharpest degree bound $2(d^{2}/2+d)^{2^{n-1}}\sim 2d^{2^{n}}$.

In 2005, Caviglia and Sbarra [8] studied upper bounds for the Castelnuovo-Mumford regularity of homogeneous ideals. Analyzing Giusti’s proof, they gave a simple proof of the upper bound $(2d)^{2^{n-2}}$ for the degree reverse lexicographic Gröbner basis of an ideal $\mathcal{I}$ in generic position (they also showed that this bound holds independent of the characteristic of $\mathbbm{k}$). Finally, Mayr and Ritscher [29], by following the tracks of Dubé [10], obtained the dimension-dependent upper bound $2(1/2d^{n-D}+d)^{2^{D-1}}$ for any reduced Gröbner basis of $\mathcal{I}$. It is worth while remarking that there are also lower bounds for the complexity: $d^{2^{m}}$ with $m=n/10-O(1)$ from the work of Mayr and Meyer [28] and $d^{2^{m}}$ where $m\sim n/2$ due to Yap [37].

In this article, we will first improve Giusti’s bound by showing that if $\mathcal{I}$ is in strongly stable position and $D>1$, then $2d^{(n-D)2^{D-1}}$ is a simultaneous upper bound for the Castelnuovo-Mumford regularity of $\mathcal{I}$ and for the maximal degree of the elements of the Gröbner basis of $\mathcal{I}$ (with respect to the degree reverse lexicographic order). Furthermore, we will sharpen the bound of Caviglia-Sbarra to $(d^{n-D}+(n-D)(d-1))^{2^{D-1}}$. We will see that neither of these bounds is always greater than the other. Finally, we will show that, if $\mathcal{I}$ is in quasi stable position and $D\leq 1$, Giusti’s bound may be replaced by $nd-n+1$ (this result was already obtained by Lazard [23] when the ideal is in generic position). In the recent work [21], we showed how many variants of stable positions – including quasi stable and strongly stable position – can be achieved via linear coordinate transformations constructed with a deterministic algorithm.

The article is organized as follows. In the next section, we give basic notations and definitions. In Sections $4,5$ and $6$ we improve the degree bounds provided by Giusti, Caviglia-Sbarra and Lazard, respectively.

Throughout this article, we keep the following notations. Let $\mathcal{P}=\mathbbm{k}[x_{1},\dots,x_{n}]$ be the polynomial ring (where $\mathbbm{k}$ is of characteristic zero). A power product of the variables $x_{1},\dots,x_{n}$ is called term and $\mathbbm{T}$ denotes the monoid of all terms in $\mathcal{P}$. We consider non-zero homogeneous polynomials $f_{1},\ldots,f_{k}\in\mathcal{P}$ and the ideal $\mathcal{I}=\langle f_{1},\ldots,f_{k}\rangle$ generated by them. We assume that $f_{i}$ is of degree $d_{i}$ and that the numbering is such that $d_{1}\geq d_{2}\geq\cdots\geq d_{k}>0$. We also set $d=d_{1}$. Furthermore, we denote by $\mathcal{R}=\mathcal{P}/\mathcal{I}$ the corresponding factor ring and by $D$ its dimension. Finally, we use throughout the degree reverse lexicographic order with $x_{n}\prec\cdots\prec x_{1}$.

The leading term of a polynomial $f\in\mathcal{P}$, denoted by ${{\rm LT}}(f)$, is the greatest term (with respect to $\prec$) appearing in $f$ and its coefficient is the leading coefficient of $f$ and we denote it by ${{\rm LC}}(f)$. The leading monomial of $f$ is the product ${{\rm LM}}(f)={{\rm LC}}(f){{\rm LT}}(f)$. The leading ideal of $\mathcal{I}$ is defined as ${{\rm LT}}(\mathcal{I})=\langle{{\rm LT}}(f)\ |\ f\in{\mathcal{I}}\rangle$. For the finite set $F=\{f_{1},\ldots,f_{k}\}\subset{\mathcal{P}}$, ${\rm LT}(F)$ denotes the set $\{{\rm LT}(f_{1}),\ldots,{\rm LT}(f_{k})\}$. A finite subset $G\subset{\mathcal{I}}$ is called a Gröbner basis of $\mathcal{I}$ w.r.t. $\prec$, if ${{\rm LT}}(\mathcal{I})=\langle{\rm LT}(G)\rangle$. We refer to [1] for more details on Gröbner bases.

Given a graded $\mathcal{P}$-module $X$ and a positive integer $s$, we denote by $X_{s}$ the set of all homogeneous elements of $X$ of degree $s$. To define the Hilbert regularity of an ideal, recall that the Hilbert function of $\mathcal{I}$ is defined by ${\rm HF}_{\mathcal{I}}(t)=\dim_{\mathbbm{k}}(\mathcal{R}_{t})$; the dimension of $\mathcal{R}_{t}$ as a $\mathbbm{k}$-linear space. From a certain degree on, this function of $t$ is equal to a polynomial in $t$, called Hilbert polynomial, and denoted by ${\rm HP}_{\mathcal{I}}$ (see [9] for more details on this topic). The Hilbert regularity of $\mathcal{I}$ is ${\rm hilb}(\mathcal{I})=\min\{m\ \arrowvert\ \forall t\geq m,\ {\rm HF}_{\mathcal{I}}(t)={\rm HP}_{\mathcal{I}}(t)\}$. Finally, recall that the Hilbert series of $\mathcal{I}$ is the power series ${{\rm HS}_{\mathcal{I}}}(t)=\sum_{s=0}^{\infty}{{{\rm HF}_{\mathcal{I}}}(s)t^{s}}$.

For a proof of this result, we refer to [13, Thm. 7, page 130]. It follows immediately from Macaulay’s theorem that the Hilbert function of $\mathcal{I}$ is the same as that of ${\rm LT}(\mathcal{I})$ and this provides an effective method to compute it using Gröbner bases, see e.g. [18].

Let us state some auxiliary results on regular sequences. Recall that a sequence of polynomials $f_{1},\ldots,f_{k}\in\mathcal{P}$ is called regular if $f_{i}$ is a non-zero divisor on the ring $\mathcal{P}/\langle f_{1},\ldots,f_{i-1}\rangle$ for $i=2,\ldots,k$. This is equivalent to the condition that $f_{i}$ does not belong to any associated prime of $\langle f_{1},\ldots,f_{i-1}\rangle$. It can be shown that the Hilbert series of a regular sequence $f_{1},\ldots,f_{k}$ is equal to $\prod_{i=1}^{k}(1-t^{d_{i}})/(1-t^{n})$, see e.g. [25]. The converse of this result is also true, see [13, Exercise 7, page 137]. In addition, these conditions are equivalent to the statement that $D=n-k$.

For more details on the regularity, we refer to [32, 12, 3, 5]. It is well-known that in generic coordinates $\operatorname{reg}(\mathcal{I})$ is an upper bound for the degree of the Gröbner basis w.r.t. the degree reverse lexicographic order. This upper bound is sharp, if the characteristic of $\mathbbm{k}$ is zero (see [3]). A good measure to estimate the complexity of the computation of the Gröbner basis of $\mathcal{I}$ is the maximal degree of the polynomials which appear in this computation (see [22, 23, 16]).

We conclude this section with a brief review of the theory of Pommaret bases. Suppose that $f\in\mathcal{P}$ and ${\rm LT}(f)=x^{\alpha}$ with $\alpha=(\alpha_{1},\ldots,\alpha_{n})$. We call $\max{\{i\mid\alpha_{i}\neq 0\}}$ the class of $f$, denoted by $\operatorname{cls}({f})$. Then the multiplicative variables of $f$ are ${\mathcal{X}}_{P}(f)=\{x_{\operatorname{cls}({f})},\ldots,x_{n}\}$. Furthermore, $x^{\beta}$ is a Pommaret divisor of $x^{\alpha}$, written $x^{\beta}\mid_{P}x^{\alpha}$, if $x^{\beta}\mid x^{\alpha}$ and $x^{\alpha-\beta}\in\mathbbm{k}[x_{\operatorname{cls}({f})},\ldots,x_{n}]$.

One can easily show that any Pommaret basis is a (generally non-reduced) Gröbner basis of the ideal it generates. The main difference between Gröbner and Pommaret bases consists of the fact that by (1) any polynomial $f\in\mathcal{I}$ has a unique involutive standard representation. If an ideal $\mathcal{I}$ possesses a Pommaret basis $\mathcal{H}$, then $\operatorname{reg}(\mathcal{I})$ equals the maximal degree of an element of $\mathcal{H}$, cf. [34, Thm. 9.2]. The main drawback of Pommaret bases is however that they do not always exist. Indeed, a given ideal possesses a finite Pommaret basis, if and only if the ideal is in quasi stable position – see [34, Prop 4.4].

In the sequel, we will use the following notations: given an ideal $\mathcal{I}$ in quasi stable position, we write $\mathcal{H}=\{h_{1},\ldots,h_{s}\}$ for its Pommaret basis. Furthermore, for each $i$ we set $m_{i}={\rm LT}(h_{i})$ and it is then easy to see that $\{m_{1},\ldots,m_{s}\}$ forms a Pommaret basis of ${\rm LT}(\mathcal{I})$.

In 1984, Giusti [16] established the upper bound $(2d)^{2^{n-1}}$ for $\deg(\mathcal{I},\prec)$ in the case that the coordinates are in generic position. The key point of Giusti’s proof is the use of the combinatorial structure of the generic initial ideal in characteristic zero. Later on, Mora [31, Ch. 38], by a deeper analysis of Giusti’s proof, improved this bound to $(d+1)^{(n-D)2^{D-\lambda}}$ where $\lambda$ is the depth of $\mathcal{I}$. In this section, we improve Mora’s bound by following his general approach and correcting some flaws in his method. Our presentation seems to be simpler than the ones by Mora and Giusti.

We first note that for a given ideal in quasi stable position, we are able to reduce the number of variables by the depth of the ideal to obtain a sharper bound for $\deg(\mathcal{I},\prec)$. A novel proof à la Pommaret of this result is given below.

To state the refined version of Giusti’s bound, we need to recall the crystallisation principle. Let $A=(a_{ij})\in\mathrm{GL}(n,\mathbbm{k})$ be an $n\times n$ invertible matrix. By $A.\mathcal{I}$ we mean the ideal generated by the polynomials $A.f$ with $f\in\mathcal{I}$ where $A.f=f(\sum_{i=1}^{n}{a_{i1}x_{i}},\ldots,\sum_{i=1}^{n}{a_{in}x_{i}})$. The following fundamental theorem is due to Galligo [14].

Suppose that $\mathcal{I}=\langle f_{1},\ldots,f_{k}\rangle$ and that for some $s\in\mathbbm{N}$ we have $\deg(f_{i})\leq s$ for all $i$ and $\operatorname{gin}(\mathcal{I})$ has no minimal generator in degree $s+1$. Then, the crystallisation principle (CP) states that for each $m$ in the generating set of $\operatorname{gin}(\mathcal{I})$ we have $\deg(m)\leq s$, see [17, Prop 2.28]. Note that this principle holds only in characteristic zero and it has been proven only for generic initial ideals and for lexicographic ideals (see [17, Thm. 3.8]).

Giusti’s proof consists in applying this property along with an induction on the number of variables. One crucial fact in this direction is that CP also holds for a generic initial ideal modulo the last variable. Below, we will show that both properties remain true for arbitrary strongly stable ideals.

As a consequence of the proof of this proposition, we can infer a generalization of CP.

In the sequel, for an index $i$ we denote by $\mathcal{I}_{i}$ the ideal $\mathcal{I}|_{x_{i}=\cdots=x_{n}=0}\subset\mathbbm{k}[x_{1},\ldots,x_{i-1}]$. Since we assume that $\prec$ is the degree reverse lexicographic term order, strongly stable position of $\mathcal{I}$ entails that $\mathcal{I}_{i}$ is in strongly stable position, too, for any index $i$. The essence of Giusti’s approach consists of finding, by repeated evaluation, relations between $\deg(\mathcal{I},\prec)$ and $\deg(\mathcal{I}_{i},\prec)$ for $i=n,\ldots,n-D+1$. For this purpose, we introduce some further notations for an ideal $\mathcal{I}$ in strongly stable position. We denote by $N(\mathcal{I})$ the set of all terms $m\notin{\rm LT}(\mathcal{I})$. If $\dim(\mathcal{I})=0$, then we define $F(\mathcal{I})=N(\mathcal{I})$. Otherwise we set $F(\mathcal{I})=\{\tau x_{n}^{a}\in N(\mathcal{I})\ |\ \tau\in F(\mathcal{I}_{n})\text{ and }\deg(\tau x_{n}^{a})<\deg(\mathcal{I},\prec)\}$. Since $\mathcal{I}$ is in strongly stable position, $N(\mathcal{I})$ is strongly stable for the reverse ordering of the variables. More precisely, if $x^{\alpha}\in N(\mathcal{I})$ with $\alpha_{i}>0$, then we claim that $x_{j}x^{\alpha}/x_{i}\in N(\mathcal{I})$ for any $j>i$. Indeed, otherwise it belonged to ${\rm LT}(\mathcal{I})$ and thus – since ${\rm LT}(\mathcal{I})$ is strongly stable – $x^{\alpha}\in{\rm LT}(\mathcal{I})$ which is a contradiction.

In the case that $\mathcal{I}$ is a zero-dimensional ideal, we can derive explicit upper bounds for $\deg(\mathcal{I},\prec)$ and $\#F(\mathcal{I})$ using the following well-known lemma. We include an elementary proof for the sake of completeness.

We state now the main result of this section.

Using Prop. 3.1, we obtain even sharper bounds depending on both the dimension and the depth of $\mathcal{I}$. We continue to write $\dim(\mathcal{I})=D$ and $\operatorname{depth}(\mathcal{I})=\lambda$. It is well-known that we always have $D\geq\lambda$ (a simple proof using Pommaret bases can be found in [34] after Prop. 3.19). If $D=\lambda$, then $\mathcal{R}$ is Cohen-Macaulay. In this case, a nearly optimal upper bound for $\deg(\mathcal{I},\prec)$ exists. Recall that a homogeneous ideal $\mathcal{I}\subset\mathcal{P}$ is in Nœther position, if the ring extension $\mathbbm{k}[x_{n-D+1},\ldots,x_{n}]\hookrightarrow\mathcal{P}/\mathcal{I}$ is integral. Alternatively, Nœther position can be defined combinatorially as a weakened version of quasi stable position (see [21, Thm. 4.4]).

For the rest of this section, we thus assume that $\mathcal{R}$ is not Cohen-Macaulay, i. e. that $D>\lambda$.