Efficient Gr\"obner Bases Computation over Principal Ideal Rings

TL;DR

This paper introduces an efficient method for computing strong Gr"obner bases over quotients of principal ideal rings, leveraging a recursive reduction to computations over fields for squarefree moduli.

Contribution

The authors present a novel lifting process that reduces Gr"obner basis computations over quotients of principal ideal rings to simpler computations over fields, improving efficiency.

Findings

Reduces complex computations to simpler field-based calculations

Allows recursive reduction for squarefree moduli

Provides a practical approach for strong Gr"obner basis computation

Abstract

In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two computations over $R/aR$ and $R/bR$ where $n = ab$ with coprime $a, b$. Possibly using available factorization algorithms we may thus recursively reduce some strong Gr\"obner basis computations to Gr\"obner basis computations over fields for prime factors of $n$, at least for squarefree $n$. Considering now a computation over $R/nR$ we can run a standard Gr\"obner basis algorithm pretending $R/nR$ to be field. If we discover a non-invertible leading coefficient $c$, we use this information to try to split $n = ab$ with coprime $a, b$. If no such $c$ is discovered, the returned Gr\"obner basis is already a strong Gr\"obner basis for the input ideal over $R/nR$.

Tables

Table 1. Table 1: Benchmark timings given in seconds, computations in ℤ n subscript ℤ 𝑛 \mathbb{Z}_{n} with n = 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 19 ⋅ 23 𝑛 ⋅ 2 3 5 7 11 13 17 19 23 n=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23

Examples	New Algorithm	Singular	Magma
Cyclic-6	$0.233$	$4.453$	$1.170$
Cyclic-7	$3.871$	$1,689.805$	$303.579$
Cyclic-8	$26.211$	$>24$h	$8,304.070$
Katsura-8	$2.152$	$89.154$	$12.980$
Katsura-9	$6.831$	$1,592.969$	$133.452$
Katsura-10	$29.814$	$21,316.400$	$2,546.010$
Eco-10	$2.468$	$67.440$	$110.530$
Eco-11	$12.329$	$738.252$	$2,799.549$
F-744	$1.131$	$61.754$	$24.810$
F-855	$4.879$	$1,685.831$	$>24$h
Noon-7	$2.309$	$47.787$	$388.050$
Noon-8	$25.398$	$2,809.623$	$>24$h
Reimer-5	$0.749$	$6.899$	$4.030$
Reimer-6	$5.748$	$869.740$	$1,282.950$
Lichtblau	$0.124$	$2.904$	$1.630$
Mayr-42 *	$32.672$	$332.307$	$199.360$
Yang-1 *	$54.851$	$210.097$	$115.130$
Jason-210	$52.167$	$178.348$	$>24$h

Table 2. Table 2: Benchmark timings given in seconds, computations in ℤ n subscript ℤ 𝑛 \mathbb{Z}_{n} with n = 32771 ⋅ 32779 𝑛 ⋅ 32771 32779 n=32771\cdot 32779

Examples	New Algorithm	Singular	Magma
Cyclic-6	$0.004$	$0.237$	$0.070$
Cyclic-7	$0.179$	$53.924$	$3.130$
Cyclic-8	$2.481$	$5,970.100$	$257.180$
Katsura-8	$0.018$	$1.252$	$0.330$
Katsura-9	$0.098$	$11.066$	$2.390$
Katsura-10	$0.613$	$113.251$	$18.960$
Eco-10	$0.153$	$15.824$	$2.970$
Eco-11	$0.909$	$201.770$	$22.360$
F-744	$0.016$	$2.182$	$0.250$
F-855	$0.079$	$11.939$	$1.710$
Noon-7	$0.171$	$3.993$	$1.140$
Noon-8	$1.234$	$78.320$	$8.920$
Reimer-5	$0.006$	$0.136$	$0.130$
Reimer-6	$0.066$	$3.036$	$2.320$
Lichtblau	$0.002$	$0.041$	$0.010$
Mayr-42	$32.168$	$309.142$	$333.880$
Yang-1	$53.562$	$193.240$	$196.930$
Jason-210	$2.781$	$12.576$	$62.040$

Table 3. Table 3: Benchmark timings given in seconds, computations in ℤ n subscript ℤ 𝑛 \mathbb{Z}_{n} with n = 3 3 ⋅ 5 3 ⋅ 7 3 ⋅ 11 3 𝑛 ⋅ superscript 3 3 superscript 5 3 superscript 7 3 superscript 11 3 n=3^{3}\cdot 5^{3}\cdot 7^{3}\cdot 11^{3}

Examples	New Algorithm	Singular	Magma
Cyclic-6	$2.754$	$5.374$	$0.720$
Cyclic-7	$679.141$	$1,638.422$	$1,209.289$
Cyclic-8	$72,386.042$	$>24$h	$9,471.340$
Katsura-8	$27.054$	$59.659$	$40.050$
Katsura-9	$433.774$	$883.497$	$599.736$
Katsura-10	$1,239.260$	$11,002.259$	$\mathrm{8.528.819}$
Eco-10	$28.426$	$66.515$	$574.949$
Eco-11	$778.855$	$1,058.806$	$>24$h
F-744	$169.179$	$343.612$	$\mathrm{28.996.159}$
F-855	$3,434.641$	$21,941.538$	$>24$h
Noon-7	$1,210.192$	$13,291.706$	$>24$h
Noon-8	$78,048.640$	$>24$h	$>24$h
Reimer-5	$2.717$	$8.061$	$27.660$
Reimer-6	$92.295$	$927.624$	$14,069.210$
Lichtblau	$0.279$	$5.079$	$4.060$
Mayr-42 *	$32.601$	$362.176$	$327.420$
Yang-1 *	$60.499$	$220.546$	$204.740$
Jason-210	$171.565$	$122.727$	$>24$h

Equations

$$\mathrm{Ann}\left(I\right)=\left\{c\in \mathcal{R}\mid c\cdot a=0\forall a\in I\right\}.$$

$$f=\sum _v{a}_v{x}^v\text{vbox}..=\sum _{v\in {\mathbb{N}}^n}^{\text{finite}}{a}_{v_1,\dots ,v_n}\prod _{i=1}^n{x}_i^{v_i},$$

$$f-\frac{\mathrm{lc}\left(f\right)}{\mathrm{lc}\left(g\right)}\frac{\mathrm{lm}\left(f\right)}{\mathrm{lm}\left(g\right)}g.$$

$$\mathrm{spoly}\left(f,g\right)=a_f{t}_{f}f-a_g{t}_{g}g.$$

$$\mathrm{gpoly}\left(f,g\right)=b_f{t}_{f}f+b_g{t}_{g}g.$$

$$\mathrm{apoly}\left(f\right)=af=a\mathrm{tail}\left(f\right).$$

$$\mathrm{gpoly}\left(2x,3x\right)=(-1)\cdot 2x-(-1)\cdot 3x=x.$$

$$\left(\mathrm{lc}\left(g\right)\cdot {\left(\mathrm{lc}\left(f\right)\right)}^{-1}\right)\mathrm{lc}\left(f\right)=\mathrm{lc}\left(g\right).$$

$$f_{k,l}=$$

$$+$$

$$(\mathrm{lc}\left(g_{a,k}\right)\mathrm{lc}\left(g_{b,l}\right))\mathcal{R}=(\mathrm{lc}\left(g_{a,k}\right))\mathcal{R}\cap (\mathrm{lc}\left(g_{b,l}\right))\mathcal{R}⊋a\mathcal{R}\cap b\mathcal{R}=\{0\}$$

$$ua\cdot \mathrm{lc}\left(g_{a,k}\right)\mathrm{lc}\left(g_{b,l}\right)+vb\cdot \mathrm{lc}\left(g_{b,k}\right)\mathrm{lc}\left(g_{a,l}\right)=\mathrm{lc}\left(g_{a,k}\right)\mathrm{lc}\left(g_{b,l}\right)\ne 0.$$

$$\mathrm{lt}\left(f_{k,l}\right)=\mathrm{lcm}\left(\mathrm{lm}\left(g_{a,k}\right),\mathrm{lm}\left(g_{b,l}\right)\right)\cdot \mathrm{lc}\left(g_{a,k}\right)\mathrm{lc}\left(g_{b,l}\right).$$

$$f_{k,l}=$$

$$+$$

Full text

1 Introduction

In $1964$ Hironaka already investigated computational approaches towards singularities and introduced the notion of standard bases for local monomial orders, see, for example, Hironaka (1964); Hironaka, H. (1964); Grauert (1972). In Buchberger (1965, 2006), Buchberger initiated, in $1965$, the theory of Gröbner bases for global monomial orders by which many fundamental problems in mathematics, science and engineering can be solved algorithmically. Specifically, he introduced some key structural theory, and based on this theory, proposed the first algorithm for computing Gröbner bases. Buchberger’s algorithm introduced the concept of critical pairs and repeatedly carries out a certain polynomial operation (called reduction).

Once the underlying structure is no longer a field, one needs the notion of strong Gröbner bases respectively strong standard bases. Influential work was done by Kandri-Rody and Kapur (1988), introducing the first generalization of Buchberger’s algorithm over Euclidean domains computing strong Gröbner bases. Since then only a few optimizations have been introduced, see, for example, Wienand (2011); Lichtblau (2012); Eder et al. (2017). For more general rings, like principal ideal domains or rings, more recent approaches can be found, for example, in Norton and Sǎlǎgean (2001); Pauer (2007); Popescu (2016); Francis and Verron (2019). Common to all these approaches is the idea to transfer ideas from the well studied field case, like criteria for predicting zero reductions or the use of linear algebra, to the setting of rings.

In some sense, we take this approach to the extreme by just treating the underlying ring as field and by hopefully splitting the computation to smaller problems in case it fails. To be more precise, consider a quotient $\mathcal{R}/n\mathcal{R}$ of a principal ideal domain $\mathcal{R}$ for some non-trivial element $n\in\mathcal{R}$. If $I\subseteq(\mathcal{R}/n\mathcal{R})[x]$ is an ideal for which we want to find a strong Gröbner basis computation, we pretend that $n$ is prime, that is, $\mathcal{R}/n\mathcal{R}$ is a field and apply a classical Gröbner basis algorithm from the field case to $I$. If this does not encounter a non-invertible element, then we are done. Otherwise we use a non-invertible element to split $n=ab$ with coprime elements $a,b\in R$. After computing strong Gröbner bases of $I$ over $\mathcal{R}/a\mathcal{R}$ and $\mathcal{R}/b\mathcal{R}$, we pull them back along the canonical isomorphism $\mathcal{R}/n\mathcal{R}\to\mathcal{R}/a\mathcal{R}\times\mathcal{R}/b\mathcal{R}$ to obtain a strong Gröbner basis of $I$. In case we cannot split $n$, we fall back to a classical algorithm for computing strong Gröbner basis. The most favorable case for the new algorithm are squarefree elements $n$, since then any non-invertible element allows us to split $n$.

The idea of working in $\mathcal{R}/n\mathcal{R}$ as if $n$ were a prime, is a common strategy in computer algebra. Other examples include the computation of matrix normal forms over $\mathcal{R}/n\mathcal{R}$, see Fieker and Hofmann (2014). To make this approach work in the setting of strong Gröbner bases, we investigate the behavior of strong Gröbner bases with respect to quotients and the Chinese remainder theorem. By properly normalizing the strong Gröbner basis, we prove that one can efficiently pull back strong Gröbner bases along a projection $\mathcal{R}\to\mathcal{R}/n\mathcal{R}$ (Theorem 10) as well as along a canonical isomorphism $\mathcal{R}/n\mathcal{R}\to\mathcal{R}/a\mathcal{R}\times\mathcal{R}/b\mathcal{R}$ (Theorem 12).

The algorithm has been implemented to compute strong Gröbner bases over residue class rings of the form $\mathbb{Z}/n\mathbb{Z}$, where $n\in\mathbb{Z}_{>0}$, see Section 5. Running standard benchmarks for Gröbner basis computations for $n$ of different shape shows a consistent speed-up across all examples (except one). In case of squarefree $n$, the new algorithm improves upon the state of the art implementations by a factor of 10–100.

Acknowledgments

This work was supported by DFG project SFB-TRR 195. The authors thank Claus Fieker for helpful comments.

2 Basic notions

Let $\mathcal{R}$ be a principal ideal ring, that is a unital commutative ring such that every ideal is principal. Note that $\mathcal{R}$ is not necessarily an integral domain. If $a,b\in\mathcal{R}$ are two elements with $a\mathcal{R}\subseteq b\mathcal{R}$ we denote by $a/b$ by abuse of notation any element $c\in\mathcal{R}$ with $c\cdot b=a$. Recall that a least common multiple of two elements $a,b$ is an element $l\in\mathcal{R}$ such that $l\mathcal{R}=a\mathcal{R}\cap b\mathcal{R}$. By abuse of notation we denote by $\operatorname{lcm}(a,b)$ such an element. Similarly, we denote by $\gcd(a,b)$ an element of $\mathcal{R}$ with $r\mathcal{R}+s\mathcal{R}=\gcd(a,b)\mathcal{R}$ and call it a greatest common divisor. For an element $n\in\mathcal{R}$ we denote by $\pi_{n}\colon\mathcal{R}\to\mathcal{R}/n\mathcal{R}$ the canonical projection. For an ideal $I\subset\mathcal{R}$ we define the annihilator of $I$ by

[TABLE]

For an element $a\in\mathcal{R}$ we denote by $\mathrm{Ann}\left({a}\right):=\mathrm{Ann}\left({\langle a\rangle}\right)$ the annihilator of $a$.

A polynomial in $n$ variables $x_{1},\ldots,x_{n}$ over $\mathcal{R}$ is a finite $\mathcal{R}$-linear combination of terms $a_{v_{1},\ldots,v_{n}}\prod_{i=1}^{n}x_{i}^{v_{i}}$,

[TABLE]

such that $v\in\mathbb{N}^{n}$ and $a_{v}\in\mathcal{R}$. The polynomial ring $\mathcal{R}[x]\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{R}[x_{1},\ldots,x_{n}]$ in $n$ variables over $\mathcal{R}$ is the set of all polynomials over $\mathcal{R}$ together with the usual addition and multiplication. For $f=\sum_{v}a_{v}x^{v}\neq 0\in\mathcal{R}[x]$ we define the degree of $f$ by $\deg(f):=\max\left\{v_{1}+\cdots+v_{n}\mid v\in\mathbb{N}^{n},a_{v}\neq 0\right\}$. For $f=0$ we set $\deg(f):=-1$.

We fix once and for all a monomial order $<$ on $\mathcal{R}[x]$, which, for the sake of simplicity, is assumed to be global, that is, $x^{\alpha}\geq 1$ for all $\alpha\in\mathbb{N}^{n}$. Given a monomial order $<$ we can highlight the maximal terms of elements in $\mathcal{R}[x]$ with respect to $<$: For $f\in\mathcal{R}[x]\ \backslash\ \{0\}$, $\operatorname{lt}\left({f}\right)$ is the lead term, $\operatorname{lm}\left({f}\right)$ the lead monomial, and $\mathrm{lc}\left({f}\right)$ the lead coefficient of $f$. For any set $F\subset\mathcal{R}[x]$ we define the lead ideal $L(F)=\langle\operatorname{lt}\left({f}\right)\mid f\in F\rangle$; for an ideal $I\subset\mathcal{R}[x]$,  $L(I)$ is defined as the ideal of lead terms of all elements of $I$.

The reduction process of two polynomials $f$ and $g$ in $\mathcal{R}[x]$ depends now on the uniqueness of the minimal remainder in the division algorithm in $\mathcal{R}$:

Definition 1**.**

Let $f,g\in\mathcal{R}[x]$ and let $G=\{g_{1},\ldots,g_{r}\}\subset\mathcal{R}[x]$ be a finite set of polynomials.

We say that $g$ top-reduces $f$ if $\operatorname{lm}\left({g}\right)\mid\operatorname{lm}\left({f}\right)$ and $\mathrm{lc}\left({g}\right)\mid\mathrm{lc}\left({f}\right)$. A top-reduction of $f$ by $g$ is then given by

[TABLE] 2. 2.

Relaxing the reduction of the lead term to any term of $f$, we say that $g$ reduces $f$. In general, we speak of a reduction of a polynomial $f$ with respect to a finite set $F\subset\mathcal{R}[x]$. Let 3. 3.

We say that $f$ has a weak standard representation with respect to $G$ if $f=\sum_{i=1}^{r}h_{i}g_{i}$ for some $h_{i}\in\mathcal{R}[x]$ such that $\operatorname{lm}\left({f}\right)=\operatorname{lm}\left({h_{j}g_{j}}\right)$ for some $j\in\{1,\ldots,r\}$. 4. 4.

We say that $f$ has a strong standard representation with respect to $G$ if $f=\sum_{i=1}^{r}h_{i}g_{i}$ for some $h_{i}\in\mathcal{R}[x]$ such that $\operatorname{lm}\left({f}\right)=\operatorname{lm}\left({h_{j}g_{j}}\right)$ for some $j\in\{1,\ldots,r\}$ and $\operatorname{lm}\left({f}\right)>\operatorname{lm}\left({h_{k}g_{k}}\right)$ for all $k\neq j$.

This kind of reduction is equivalent to definition CP3 from Kandri-Rody and Kapur (1984) and generalizes Buchberger’s attempt from Buchberger (1985). The result of such a reduction might not be unique. This uniqueness is exactly the property Gröbner bases give us.

Definition 2**.**

A finite set $G\subset\mathcal{R}[x]$ is called a Gröbner basis for an ideal $I$ (with respect to $<$) if $G\subset I$ and $L(G)=L(I)$. Furthermore, $G$ is called a strong Gröbner basis if for any $f\in I\backslash\{0\}$ there exists an element $g\in G$ such that $\operatorname{lt}\left({g}\right)\mid\operatorname{lt}\left({f}\right)$.

Remark 3**.**

Note that $G$ being a strong Gröbner basis is equivalent to all elements $g\in G$ having a strong standard representation with respect to $G$. See, for example, Theorem 1 in Lichtblau (2012) for a proof.

Clearly, assuming that $\mathcal{R}$ is a field, any Gröbner basis is a strong Gröbner basis. But in our setting with $\mathcal{R}$ being a principal ideal ring one has to check the coefficients, too, as explained in Definition 1. The fact that for an arbitrary principal ideal ring the notions of Gröbner bases and strong Gröbner bases do not agree, can be observed already for monomial ideals in univariate polynomial rings:

Example 4**.**

Let $\mathcal{R}=\mathbb{Z}$ and $I=\langle x\rangle\in\mathcal{R}[x]$. Clearly, $G:=\{2x,3x\}$ is a Gröbner basis for $I$: $L(I)=\langle x\rangle$ and $x=3x-2x\in L(G)$. But $G$ is not a strong Gröbner basis for $I$ since $2x\nmid x$ and $3x\nmid x$.

In order to compute strong Gröbner bases we need to consider two different types of special polynomials:

Definition 5**.**

Let $f,g\in\mathcal{R}[x]$, $t=\operatorname{lcm}\left(\operatorname{lm}\left({f}\right),\operatorname{lm}\left({g}\right)\right)$, $t_{f}=\frac{t}{\operatorname{lm}\left({f}\right)}$, and $t_{g}=\frac{t}{\operatorname{lm}\left({g}\right)}$.

Let $a=\operatorname{lcm}\left(\mathrm{lc}\left({f}\right),\mathrm{lc}\left({g}\right)\right)$, $a_{f}=\frac{a}{\mathrm{lc}\left({g}\right)},$ and $a_{g}=\frac{a}{\mathrm{lc}\left({f}\right)}$. A S-polynomial of $f$ and $g$ is denoted by

[TABLE] 2. 2.

Let $b=\gcd\left(\mathrm{lc}\left({f}\right),\mathrm{lc}\left({g}\right)\right)$. Choose $b_{f},b_{g}\in\mathcal{R}$ such that $b=b_{f}\mathrm{lc}\left({f}\right)+b_{g}\mathrm{lc}\left({g}\right)$. A GCD-polynomial of $f$ and $g$ is denoted by

[TABLE] 3. 3.

Let $a\in\mathcal{R}$ be a generator of $\mathrm{Ann}\left({\mathrm{lc}\left({f}\right)}\right)$. An annihilator polynomial of $f$ is denoted by

[TABLE]

Remark 6**.**

Note that $\mathrm{spoly}\left({f},{g}\right),\mathrm{gpoly}\left({f},{g}\right)$ as well as $\mathrm{apoly}\left({f}\right)g$ are not uniquely defined, since quotients, Bézout coefficients and generators are in general not unique. 2. 2.

If $\mathrm{lc}\left({f}\right)$ is not a zero divisor in $\mathcal{R}$ then $\mathrm{apoly}\left({f}\right)=0$. It follows that if $\mathcal{R}$ is a domain, there is no need to handle annihilator polynomials since [math] is the only zero divisor. 3. 3.

*In the field case we do not need to consider GCD-polynomials at all since we can always normalize the polynomials, that is, ensure that $\mathrm{lc}\left({f}\right)=1$. *

From Example 4 it is clear that the usual Buchberger algorithm as in the field case will not compute a strong Gröbner basis as we would only consider $\mathrm{spoly}\left({2x},{3x}\right)=3\cdot 2x-2\cdot 3x=0$. Luckily, we can fix this via taking care of the corresponding GCD-polynomial:

[TABLE]

It follows that given an ideal $I\subset\mathcal{R}[x]$ a strong Gröbner basis for $I$ can be achieved using a generalized version of Buchberger’s algorithm computing not only strong standard representations of S-polynomials but also of GCD-polynomials and annihilator polynomials. We refer, for example, to Lichtblau (2012) for more details.

So, how do we get a strong standard representations of elements w.r.t. some set $G$? The answer is given by the concept of a normal form:

Definition 7**.**

Let $\mathcal{G}$ denote the set of all finite subsets $G\subset\mathcal{R}[x]$. We call the map $\text{{NF}}:\mathcal{R}[x]\times\mathcal{G}\longrightarrow\mathcal{R}[x],\,(f,G)\longmapsto\text{{NF}}\left({f},{G}\right)$, a weak normal form (w.r.t. a monomial ordering $<$) if for all $f\in\mathcal{R}[x]$ and all $G\in\mathcal{G}$ the following hold:

$\text{{NF}}\left({0},{G}\right)=0$. 2. 2.

If $\text{{NF}}\left({f},{G}\right)\neq 0$ then $\operatorname{lt}\left({\text{{NF}}\left({f},{G}\right)}\right)\notin L(G)$. 3. 3.

If $f\neq 0$ then there exists a unit $u\in\mathcal{R}[x]$ such that either $uf=\text{{NF}}\left({f},{G}\right)$ or $r=uf-\text{{NF}}\left({f},{G}\right)$ has a strong standard representation with respect to $G$.

A weak normal form NF is called a normal form if we can always choose $u=1$.

Algorithm 1 presents a normal form algorithm for computations:

Now we state Buchberger’s algorithm for computing strong Gröbner bases, Algorithm 2. For the theoretical background we refer to Greuel and Pfister (2007) and Becker and Weispfenning (1993).

Algorithm 2 Buchberger’s algorithm for computing strong Gröbner bases (sBBA)

1:Ideal $I=\langle f_{1},\ldots,f_{m}\rangle\subset\mathcal{R}[x]$, normal form algorithm NF (depending on $<$)

2:Gröbner basis $G$ for $I$ w.r.t. $<$

3:$G\leftarrow\{f_{1},\ldots,f_{m}\}$

4:$P\leftarrow\left\{\mathrm{apoly}\left({f_{i}}\right)\mid 1\leq i\leq n\right\}$

5:$P\leftarrow\left\{\mathrm{spoly}\left({f_{i}},{f_{j}}\right),\mathrm{gpoly}\left({f_{i}},{f_{j}}\right)\mid 1\leq i<j\leq m\right\}$

6:while $\left(P\neq\emptyset\right)$ do

7:     Choose $h\in P$, $P\leftarrow P\setminus\{h\}$

8:     $h\leftarrow\text{{NF}}\left({h},{G}\right)$

9:     if $\left(h\neq 0\right)$ then

10:         $P\leftarrow P\cup\left\{\mathrm{apoly}\left({h}\right)\right\}$

11:         $P\leftarrow P\cup\left\{\mathrm{spoly}\left({g},{h}\right),\mathrm{gpoly}\left({g},{h}\right)\mid g\in G\right\}$

12:         $G\leftarrow G\cup\{h\}$      

13:$\text{{return }}G$

3 Strong Gröbner bases over principal ideal rings

In this section we give theoretical results for the computation of strong Gröbner bases over principal ideal rings. These results will then be used in Section 4 for an improved computation of strong Gröbner bases over quotients of principal ideal rings. We begin by analyzing Algorithm 2 in case all occurring leading coefficients are invertible.

Lemma 8**.**

Let $I=\langle f_{1},\ldots,f_{m}\rangle\subset\mathcal{R}[x]$ be an ideal such that for all $i=1,\ldots,m$ we have that $\mathrm{lc}\left({f_{i}}\right)$ is invertible in $\mathcal{R}$. Moreover, assume that for each newly added polynomial $h$ in Line 12 in Algorithm 2 the polynomial $\mathrm{lc}\left({h}\right)$ is invertible in $\mathcal{R}$. Then Algorithm 2 does not need to consider GCD-polynomials and annihilator polynomials.

Proof.

We show that all GCD-polynomials and all annihilator polynomials are zero in the setting of the lemma:

For each element $g$ in the intermediate Gröbner basis $G$ it holds that $\mathrm{lc}\left({g}\right)$ is invertible in $\mathcal{R}$ and thus, not a zero divisor. It follows that $\mathrm{apoly}\left({g}\right)=0$ by definition. 2. 2.

For each $\mathrm{gpoly}\left({f},{g}\right)$ for $f,g\in G$ it holds that $\mathrm{lc}\left({f}\right)\mid\mathrm{lc}\left({g}\right)$: $\mathrm{lc}\left({f}\right)$ is invertible in $\mathcal{R}$, so we get

[TABLE]

Again, by definition, $\mathrm{gpoly}\left({f},{g}\right)=0$.∎

Remark 9**.**

From Lemma 8 it follows that as long as Algorithm 2 does not encounter a lead coefficient that is not invertible in $\mathcal{R}$ we can use Buchberger’s algorithm from the field case without the need to consider GCD-polynomials and annihilator polynomials for strongness properties. In Section 4 we discuss how one can use this fact to improve the general computations of strong Gröbner bases over $\mathcal{R}[x]$.

We next next show how to pull back a strong Gröbner bases along a canonical projection $\mathcal{R}[x]\to(\mathcal{R}/n\mathcal{R})[x]$.

Theorem 10**.**

Let $n\in\mathcal{R}$, $n\neq 0$ and $I\subseteq\mathcal{R}[x]$ an ideal. Assume that $G_{n}\subseteq\mathcal{R}[x]$ is a set of polynomials with the following properties:

$\pi_{n}(G_{n})$* is a strong Gröbner basis of $\pi_{n}(I)$;* 2. 2.

for every $g\in G_{n}$ the leading coefficient $\mathrm{lc}\left({g}\right)$ divides $n$ and $\mathrm{lc}\left({g}\right)\not\in n\mathcal{R}$.

Then $G_{n}\cup\{n\}$ is a strong Gröbner basis of $I+n\mathcal{R}[x]$.

Proof.

It is clear that $G_{n}\cup\{n\}\subseteq I+n\mathcal{R}[x]$. Now let $f\in I$. If $\pi_{n}(\mathrm{lc}\left({f}\right))=0$, then the leading term is a multiple of $n$. Thus we may assume that $\pi_{n}(f)\neq 0$. Since $\pi_{n}(G_{n})$ is a strong Gröbner basis of $\pi_{n}(I)$, there exists $g\in G_{n}$ such that $\operatorname{lt}\left({\pi_{n}(g)}\right)$ divides $\operatorname{lt}\left({\pi_{n}(f)}\right)$. Hence we can find $h\in\mathcal{R}[x]$ with $\pi_{n}(h)\cdot\operatorname{lt}\left({\pi_{n}(g)}\right)=\operatorname{lt}\left({\pi_{n}(f)}\right)$. We can assume that $h$ is a term and $\operatorname{lm}\left({h}\right)\cdot\operatorname{lm}\left({g}\right)=\operatorname{lm}\left({f}\right)$. By assumption we have $\pi_{n}(\operatorname{lt}\left({g}\right))=\operatorname{lt}\left({\pi_{n}(g)}\right)$ as well as $\pi_{n}(\operatorname{lt}\left({f}\right))=\operatorname{lt}\left({\pi_{n}(f)}\right)$. Hence we can write $h\cdot\operatorname{lt}\left({g}\right)-\operatorname{lt}\left({f}\right)=c\cdot x^{\deg(f)}$ for some $c\in n\mathcal{R}$. Since $\mathrm{lc}\left({g}\right)$ divides $n$ and $\operatorname{lm}\left({g}\right)$ divides $\operatorname{lm}\left({f}\right)$ it follows that $\operatorname{lt}\left({g}\right)$ divides $\operatorname{lt}\left({f}\right)$. ∎

Remark 11**.**

Assume that we know that an ideal $I\subseteq\mathcal{R}[x]$ contains a constant polynomial $n\in\mathcal{R}$, $n\neq 0$. As $I=I+n\mathcal{R}[x]$, Theorem 10 implies that we can compute a strong Gröbner basis of $I$ be properly choosing the lifts of a strong Gröbner basis of the reduction $\pi_{n}(I)\subseteq(\mathcal{R}/n\mathcal{R})[x]$. For $\mathcal{R}=\mathbb{Z}$, a similar idea can be found in Section $4$ of Eder, C. et al. (2018). There it is described how to check if an ideal $I\subseteq\mathbb{Z}[x]$ contains a constant polynomial $n\in\mathbb{Z}$, $n\neq 0$. In case it exists, the authors describe an ad hoc method which keeps the size of the coefficients of the polynomials in Algorithm 2 bounded by $n$.

We now consider the following situation. Assume that $a,b,u,v\in\mathcal{R}$ are elements with $1=ua+vb$ and $ab=0$. Note that this implies $(ua)^{2}=ua$ and $(vb)^{2}=vb$.

Theorem 12**.**

Assume that $I\subseteq\mathcal{R}[x]$ is an ideal. Furthermore let $\operatorname{\mathcal{K}},\operatorname{\mathcal{L}}$ be finite index sets and $G_{a}=(g_{a,k})_{k\in\operatorname{\mathcal{K}}},G_{b}=(g_{b,l})_{l\in\operatorname{\mathcal{L}}}$ strong Gröbner bases of $I+a\mathcal{R}[x]$ and $I+b\mathcal{R}[x]$ respectively, satisfying the following conditions:

For $k\in\operatorname{\mathcal{K}}$, if $g_{a,k}$ is non-constant, then $\mathrm{lc}\left({g_{a,k}}\right)$ divides $a$ and $\mathrm{lc}\left({g_{a,k}}\right)\not\in a\mathcal{R}$ 2. 2.

For $l\in\operatorname{\mathcal{L}}$, if $g_{b,l}$ is non-constant, then $\mathrm{lc}\left({g_{b,l}}\right)$ divides $b$ and $\mathrm{lc}\left({g_{b,l}}\right)\not\in b\mathcal{R}$.

For $k\in\operatorname{\mathcal{K}}$, $l\in\operatorname{\mathcal{L}}$ define

[TABLE]

Then $G=\{f_{k,l}\mid k\in\operatorname{\mathcal{K}},l\in\operatorname{\mathcal{L}}\}$ is a strong Gröbner basis of $I$.

Proof.

Note that from $1=ua+vb$ it follows at once that $I=eaI+vbI$. Hence $I=eaI+vbI=ea(I+b\mathcal{R}[x])+vb(I+a\mathcal{R}[x])=ua\langle G_{b}\rangle+vb\langle G_{a}\rangle$. As $\mathrm{lc}\left({g_{a,k}}\right)$ and $\mathrm{lc}\left({g_{b,l}}\right)$ are coprime we have $\operatorname{lcm}(\mathrm{lc}\left({g_{a,k}}\right),\mathrm{lc}\left({g_{b,l}}\right))=\mathrm{lc}\left({g_{a,k}}\right)\mathrm{lc}\left({g_{b,l}}\right)$. Moreover, since

[TABLE]

we have

[TABLE]

In particular

[TABLE]

Consider now an element $h\in I$. Since $h\in I+a\mathcal{R}[x]$ and $h\in I+b\mathcal{R}[x]$, there exist $g_{a,k}\in G_{a}$, $g_{b,l}\in G_{b}$ such that $\operatorname{lt}\left({g_{a,k}}\right)\mid\operatorname{lt}\left({h}\right)$ and $\operatorname{lt}\left({g_{b,l}}\right)\mid\operatorname{lt}\left({h}\right)$. Thus $\operatorname{lm}\left({h}\right)$ is divisible by $\operatorname{lcm}\left(\operatorname{lm}\left({g_{a,k}}\right),\operatorname{lm}\left({g_{b,l}}\right)\right)$ and $\mathrm{lc}\left({h}\right)$ is divisible by $\mathrm{lc}\left({g_{a,k}}\right)\mathrm{lc}\left({g_{b,l}}\right)$, that is, $\operatorname{lt}\left({h}\right)$ is divisible by $\operatorname{lt}\left({f_{k,l}}\right)$. ∎

4 Algorithmic approach for computing in quotients of principal ideal rings

We now assume that $\mathcal{R}$ is a principal ideal domain. Additionally we now also fix an element $n\in\mathcal{R}$, $n\neq 0$. Using the theoretical results from Section 3 we are now able to describe improvements to the Gröbner basis computation over the base ring $\mathcal{R}/n\mathcal{R}$.

Corollary 13**.**

Let $I\subseteq(\mathcal{R}/n\mathcal{R})[x]$ be an ideal and $n=a\cdot b$ a factorization of $n$ into coprime elements $a,b\in\mathcal{R}$. Let $\operatorname{\mathcal{K}},\operatorname{\mathcal{L}}$ be finite index sets. Assume that $G_{a}=(g_{a,k})_{k\in\operatorname{\mathcal{K}}}\subseteq(\mathcal{R}/n\mathcal{R})[x]$ is a set of polynomials, such that $\pi_{a}(G_{a})\subseteq(\mathcal{R}/a\mathcal{R})[x]$ is a strong Gröbner basis of $\pi_{a}(I)\subseteq(\mathcal{R}/a\mathcal{R})[x]$, $a\in G_{a}$ and for every $g\in G_{a}\setminus\{a\}$, the leading coefficient $\mathrm{lc}\left({g}\right)$ divides $a$ and is not divisible by $a$. Assume that $G_{b}=(g_{b,l})_{l\in\operatorname{\mathcal{L}}}\subseteq\mathcal{R}/n\mathcal{R}[x]$ has similar properties with respect to $b$. For $i\in I$, $j\in J$ define

[TABLE]

Then $G=\{f_{k,l}\mid k\in\operatorname{\mathcal{K}},l\in\operatorname{\mathcal{L}}\}$ is a strong Gröbner basis of $I$.

Proof.

Follows at once from Theorems 10 and 12 since for $\bar{a}\in\bar{}\mathcal{R}=\mathcal{R}/n\mathcal{R}$ we have $\bar{}\mathcal{R}/\bar{a}\bar{}\mathcal{R}\cong\mathcal{R}/a\mathcal{R}$. ∎

To use this, we need, given a divisor $a\in\mathcal{R}$ of $n$, a way to lift polynomials from $\mathcal{R}/a\mathcal{R}$ to $\mathcal{R}/n\mathcal{R}$ such that the leading coefficients divide $n$.

Lemma 14**.**

There exists an algorithm, that given $c\in\mathcal{R}$ determines $u\in\mathcal{R}$ such that $\gcd(u,n)\in\mathcal{R}^{\times}$ and $uc=\gcd(c,n)\bmod n$.

Proof.

This can be found in (Storjohann and Mulders, 1998, Section 2). ∎

In case we have an algorithm for factoring elements of $\mathcal{R}$ into irreducible elements, this allows us to reduce the strong Gröbner basis computation to computations over smaller quotient rings. This approach is summarized in Algorithm 3.

Depending on the ring $\mathcal{R}$, the particular $n$ and the factorization algorithm, Step (1) is infeasible or not. For example if $\mathcal{R}=\mathbb{Z}$, the fastest factorization algorithms are subexponential in $n$, rendering this approach futile for non-trivial example with large $n$. On the other hand, if $\mathcal{R}=\mathbb{Q}[t]$ or $\mathcal{R}=\mathbb{F}_{p}[t]$ for some prime $p$, then factoring in $\mathcal{R}$ can be done in (randomized) polynomial time in the size of $n$ and thus is a good idea, at least from a theoretical point of view.

We now consider the case, where we cannot or do not want to factor the modulus $n$. The basic idea is to run the algorithm from the field case, pretending that $\mathcal{R}/n\mathcal{R}$ is a field, and to stop whenever we find a non-invertible leading coefficient. If the algorithm discovers a non-invertible element, we try to split the modulus and the computation of the strong Gröbner basis. The splitting is based on the following consequence of so-called factor refinement.

Proposition 15**.**

There exists an algorithm that given $a\in\mathcal{R}$ with $\gcd(a,n)\not\in\mathcal{R}^{\times}\cup n\mathcal{R}$, that is, $0\neq\bar{a}\notin(\mathcal{R}/n\mathcal{R})^{\times}$, either

finds $m\in\mathcal{R}$, $k\in\mathbb{Z}_{>1}$ with $n=m^{k}$ and $m=\gcd(a,m)$, or 2. 2.

finds coprime elements $p,q\in\mathcal{R}\setminus\mathcal{R}^{\times}$ with $n=p\cdot q$.

Proof.

Using an algorithm for factor refinement, for example the algorithm of Bach–Driscoll–Shallit (see Bach et al. (1993)), we can find a set $S\subseteq\mathcal{R}\setminus\mathcal{R}^{\times}$ of coprime elements such that $a$ and $n$ factor uniquely into elements of $S$ and for all $s\in S$ we have $\gcd(a,s)$ or $\gcd(b,s)$ not in $\mathcal{R}^{\times}$. Now pick $m\in S$ with $m\mid n$. We can write $n=m^{k}\cdot u$ with $k\in\mathbb{Z}_{>0}$ and $u\in\mathcal{R}$ coprime to $m$. Depending on whether $u$ is a unit or not, we are in case (1) or (2). ∎

Incorporating this into the strong Gröbner basis computation we obtain Algorithm 4.

Theorem 16**.**

Algorithm 4 terminates and is correct.

Proof.

Termination follows since in the recursion, the number of irreducible factors of $n$ is strictly decreasing. Correctness follows from Corollary 13. ∎

Remark 17**.**

The usefulness of the splitting depends very much on the factorization of $n$. For example, if $n=p^{e}$ is the power of an irreducible element $p$, then Algorithm 4 is the same as Algorithm 2. The most favorable input for Algorithm 4 are rings $\mathcal{R}/n\mathcal{R}$ with $n$ squarefree, that is, $n=p_{1}\dotsc p_{r}$ is the product of pairwise coprime irreducible elements (including the case, where $n=p_{1}$ is itself irreducible). In this case, every non-invertible element allows us to split the modulus into coprime elements. Thus all Gröbner basis computations can be done as in the field case.

5 Experimental results

In the following we present experimental results comparing our new approach to the current implementations in the computer algebra systems Singular (Decker et al. (2019)) and Magma (Bosma et al. (1997)). All computations were done on an Intel® Xeon® CPU E5-2643 v3 @ 3.40GHz with $384$GB RAM. Computations that took more than 24 hours were terminated by hand.

Our new algorithm is implemented in the Julia package GB.jl (Eder and Hofmann (2019)) which is part of the OSCAR project of the SFB TRR-195. The package GB.jl is based on the C library GB (Eder (2019)) which implements Faugère’s F4 algorithm (Faugère (1999)) for computing Gröbner bases over finite fields. The implementation of Algorithm 4 uses GB and Singular as follows: All the computations over a ring $\mathcal{R}/n\mathcal{R}$ for which we want to execute the algorithm from the field case are delegated to GB. In case we find $n\in R$ such that $n$ is not prime but we cannot find a factorization of $n$ into coprime elements (Proposition 15 (1)), we delegate the corresponding strong Gröbner basis computation to Singular. All the lifting and recombination steps are done in Julia (see Fieker et al. (2017)). Note that in practice we always compute minimal strong Gröbner bases and make sure that minimality is preserved during the recombination using Corollary 13. This makes sure that for all intermediate Gröbner bases that we compute the size is bounded by the size of a minimal strong Gröbner basis of the input.

We use a set of different benchmark systems focusing on pair handling, the reduction process, finding of reducers, respectively. We have computed strong Gröbner bases for these systems over $\mathbb{Z}_{n}$ using three different settings for $n$:

For $n=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23$ (Table 1) we get a factorization down to the finite prime fields. Thus in all theses examples our new implementation can use the F4 algorithm implemented in GB as base case. With “*” we highlight examples for which there was no non-invertible element discovered, that is, the computation from the field case runs through without any splitting of $n$ to be considered. 2. 2.

For $n=32771\cdot 32779$ (Table 2) we can see that our approach of applying Lemma 8 is very promising: In none of the examples tested we found non-invertible elements, thus we compute the basis as if we are working over a finite field, receiving a correct strong Gröbner basis over $\mathbb{Z}_{n}$. 3. 3.

For $n=3^{3}\cdot 5^{3}\cdot 7^{3}\cdot 11^{3}$ (Table 3) we, in general, have to use Singular’s strong Gröbner basis algorithm for computing in $\mathbb{Z}_{p^{k}}$. Still, we can see that our approach is most often by a factor of at least $3$ faster than directly applying Singular’s implementation over $\mathbb{Z}_{n}$. The only exception is Jason-210, for which Singular is faster: The basis is huge ($>2,009$ generators), thus our new implementation needs roughly $75$ of the overall $171$ seconds to apply the recombination and lifting due to Corollary 13. Again, we highlight with “*” examples for which there was no non-invertible element discovered.

6 Conclusion

We have presented a new approach for computing strong Gröbner bases over principal ideal rings $\mathcal{R}$ which exploits the factorization of composite moduli $n$ to recursively compute strong Gröbner bases in smaller rings and lifting the results back to $\mathcal{R}$. In many situations the base cases of this recursive step boil down to computations over finite fields which are much faster than those over principal ideal rings.

One further optimization of our new approach might be the following: Once we have several factors of $n$ found, we can run the different, independent Gröbner basis computations in parallel. This is one of our next steps. Another one is to implement an optimized version of Faugère’s F4 algorithm for $\mathcal{R}/p^{k}\mathcal{R}$ in GB.