On lexicographic Gröbner bases of radical ideals in dimension zero: interpolation and structure
Due to the elimination property held by the lexicographic monomial order, the corresponding Gröbner bases display strong structural properties from which meaningful informations can easily be extracted. We study these properties for radical ideals of (co)dimension zero. The proof presented relies on a combinatorial decomposition of the finite set of points whereby iterated Lagrange interpolation formulas permit to reconstruct a minimal Gröbner basis. This is the first fully explicit interpolation formula for polynomials forming a lexicographic Gröbner basis, from which the structure property can easily be read off. The inductive nature of the proof also yield a triangular decomposition algorithm from the Gröbner basis.
keywords:Gröbner basis, Lexicographic order, Interpolation
The lexicographic monomial order benefits fully of the elimination property, implying a lot of structure in the polynomials of such Gröbner bases. Concretely, for “non-generic” Gröbner bases, some common factors are repeated among several polynomials, inducing some redundancies. “Non-generic” implies here Gröbner bases which have more polynomials than the number of variables. These redundancies explain why they are often huge, quite impracticable for substantial computations (as compared to the degree reverse lexicographic order in particular). On the other hand, this structure makes easy the extraction of meaningful informations. For instance, an early application was the possibility to solve polynomial systems (Bu85, , Method 6.10) Ka87 () in the case where the number of solutions is finite (that is when the generated ideal is of (co)dimension zero, as we will rather say hereafter). The structure is also useful to express the polynomials in such Gröbner bases with interpolation formulas, in function of the solution points. Such formulas allow to get reasonably sharp upper bounds on the size of coefficients, as it was achieved for some special cases of lexicographic Gröbner bases in DaKaSc12 (); DaSc04 (); Da09 (). Another application is the possibility to decompose such “non-generic” Gröbner bases into smaller ones. This principle fits the realm of triangular decompositions, and the fact that starting to decompose from a lex. G.b. is easier is due to Lazard (La92, , Section 5). He sketched two methods to perform this decomposition, and claimed correctness resorting to Gianni-Kalkbrener’s theorem Ka87 (); Gi87 (). But with no more details, and it appears not obvious whether this is sufficient. Probably not, indeed a proof becomes easy when resorting to the stronger Theorem (structure) provided here. More details on this is found in the paragraph Specialization… below.
Let be a field, the polynomial ring with variables on which is put the lexicographic monomial order for which and a radical ideal of dimension zero (its associated prime ideals are all maximal). In this case the degree of is the finite integer equal to .
|(H)||Assume that is infinite, or else that . Moreover, if is of finite characteristic, for each associated prime ideal to , the finite field extension is separable (this is always the case if is a finite field).|
Let be the set of common zeroes of the polynomials in , with coordinates taken in the algebraic closure of . Because of the separability assumption (H) above, the cardinal of is equal to .
Let be a lexicographic Gröbner basis of . We assume that it is minimal, that is is a minimal monomial basis for the monomial ideal . But we do not necessarily assume that is reduced. Since the polynomial ring is over a field , it does not matter to require to be monic: all the polynomials in have a leading coefficient equal to 1. For , let . Given a polynomial , let be the leading coefficient of . Furthermore, and will denote respectively the leading term and the leading monomial of yielding the equality .
Theorem (Structure). For all , holds:
This structure theorem has a direct application in the context of specialization of Gröbner bases. The proof of this theorem is easily reduced to show the existence of one Gröbner basis which verifies these properties.
Theorem (Interpolation). Let denotes the set of common zeroes of the polynomials in . There is a combinatorial decomposition of which allows to describe each polynomial as explicit interpolation formulas (Corollary 2 and Equation (28)).
The structure theorem already appear in a work of Marinari-Mora MaMo03 (). They even managed to generalize it to slightly more general ideals than radical ones in MaMo06 (). Note that the formulation given therein is slightly different, but the above is more handy. However, some novelty is brought in, as detailed hereunder:
The proof in MaMo03 (); MaMo06 () is quite unwieldy, making it difficult to check correctness. It is build upon the combinatorial algorithm of Cerlienco-Mureddu CeMu95 () to deduce the leading monomials of minimal Gröbner bases of . More recently a more suited combinatorial algorithm “lex game” FeRaRo06 () has appeared, which is also more efficient (see discussion in § 3 therein). Our presentation of the decomposition algorithm in § 3 is quite similar, but is simplified and the proof is different and more rigorous.
based on the decomposition of § 3 we give new explicit interpolation formulas. These are well-suited to generalize the use of fast interpolation algorithms, and to derive a good running-time for the reconstruction algorithm. These explicit formulas are also necessary to obtain complexity estimates on the size of coefficients by using height theory as done in DaSc04 (); Da09 ().
a main new ingredient of the present article is the recursive point of view of the proof of Theorem 3. Besides its conceptual simplicity regarding the other previous works, it prepares the ground for a first proof of the algorithm lextriangular (mentioned above).
As already said, there is a variety of previous works dealing more or less closely with the same kind of results. A comparison with MaMo03 (); MaMo06 () has been discussed in 1. above. On the more specific part concerning interpolation there are also several previous works. Let us mention the most recent one Le08 (). It describes an algorithm to compute the reduced Gröbner basis of the ideal of vanishing polynomials on a Zariski-closed finite set. It claims (no discussion about running-times is provided) to subsume the earlier but more general Buchberger-Möller algorithm BuMo82 (). The highlight of his proof is the use of an operation on standard monomials, with well-suited Lagrange interpolation. It is therefore rigorous but no explicit interpolation formula is supplied and thus deriving the structure theorem is not obvious. Indeed, it is thanks to the explicit formulas that the proof of Theorem (Structure) is quite straightforward. On the other hand, in order to describe these formulas, we introduce a combinatorial decomposition of the zero-set which is quite technical to define.
This combinatorial decomposition is very similar to the one appearing in the “lex game” FeRaRo06 (). The purpose therein is to find the set of standard monomials of a finite set of points in the -affine space. The present work is going further with the addition of interpolation to describe the Gröbner basis, while in the same purely combinatorial manner only the set of standard monomials can also be deduced. The recursive proof given is also different and has the advantage to prepare the ground for the lextriangularalgorithm. Moreover, we have tried to reduce the number of notations as low as possible and to present the decomposition as plain as possible. This was thought in the hope to provide with this more suited decomposition than the Cerlienco-Mureddu CeMu95 (), a more rigorous and easy to read proof of the Theorem (Structure) than MaMo03 (); MaMo06 (), which is also a contribution of this work.
In comparison with Le08 (); FeRaRo06 (), the present article provides a similar combinatorial decomposition of § 3 as the one provided by the “lex-game” (see § 2.3 therein) but has the additional benefits mentioned in 2.-3. above, which are necessary to pursue the works on complexity and on lextriangular. Moreover, the interpolation part of § 4 is more explicit, with the possibility to provide a good running-time, improving upon possibly the algorithm of Le08 ().
Concerning the structure theorem, before this theorem was stated in MaMo03 (), previous works on the structure have been considered. In the easy case of a polynomial ring with two variables, Lazard has found out that the structure theorem hold for any ideal, not only radical one Laz85 ().
Theorem (D. Lazard). Let be a zero-dimensional ideal, and a minimal lexicographic Gröbner basis of for . Then:
It follows easily a factorization property of the polynomials in such a Gröbner basis, which is actually the original statement222However, the formulation above is more compact and handy, and is equivalent assuming that we are considering only minimal and monic Gröbner bases. of Lazard (Laz85, , Theorem 1 (i)). In the case of a radical ideal this is equivalent to Theorem (Structure) aforementioned. Note that if is not radical, Theorem (Structure) does not hold for .
Application to stability of Gröbner bases under specialization
The stability of a Gröbner basis under a homomorphism map goes beyond the scope of this article (see BaGaSt93 () for details). When the homomorphism map is a specialization and takes the following form: for and ,
then given a monomial order that eliminates the variables , is said to be stable under if and only if: (only the inclusion is not automatically satisfied). Hence, if is a Gröbner basis of this implies that is a Gröbner basis of . It may happen though that is a Gröbner basis of without verifying (Gi87, , Rmk 2, Ex. 3). If the monomial order is lexicographic and is radical of dimension zero as in Hypothesis (H), then Theorem (Structure) applies and implies stability. More precisely:
With the notations and assumptions above, given a minimal Göbner basis of , and , the following equivalence holds:
In particular, and the stability property holds. Hence is a Gröbner basis of .
It is noteworthy that Becker in Be94 () proves that remains a Gröbner basis but he does not prove stability333his proof can not be adapted. In the proof of the crucial Lemma 1, in Equality (4) is assumed a degree decrease that prevents to consider stability., letting unproved the fact that .
When all the variables but the largest are specialized, that is when , Gianni-Kalkbrener Gi87 (); Ka87 () has proved that in the case of a radical ideal , the stability property holds. It strongly relies on Lemma 5.6 of GTZ88 (). As shows Corollary 1, Theorem (Structure) is the genuine generalization of Gianni-Kalkbrener (which is not the result of Becker Be94 ()).
Organization of the paper
In § 2 hereunder we treat the case to keep up a geometric intuition. The other sections § 3-4-5 aim at generalizing to more than 3 variables. § 3 introduces the combinatorial decomposition of the set of points that determines the leading terms of the minimal Gröbner basis. § 4-5 are proving this fact by constructing explicitly polynomials by interpolation (§ 4), which are proved in § 5 to form a Gröbner basis.
2 Warming-up: case of three variables
The first case for which difficulties arise is when . The case is too special to reveal any genuine technical problem. Though, the way to combine a decomposition of the zero set and Lagrange interpolation formula appears in (Da09, , § 2.2)444as a matter of facts, the use of the equiprojectable decomposition becomes obsolete for which we are trying to generalize here to the case of three variables.
The notations hereunder are not restricted to the case and will be used all along the paper.
Given an -uplet and an let be the projection that forgets the last coordinates: . And let be the -th coordinate function. Given another integer such that , define . Fibers of projection maps are used intensively and it is convenient to precise the length of the starting sequence: denotes the projection , The point of this notation is when taking reciprocal images to know the dimension of the starting space.
Given a set , and for we introduce a convenient notation :
The proofs in this paper deal a lot with minimal bases of monomial ideals for a which a special notation is necessary:
Given an ideal , will denote the minimal basis of the monomial ideal .
If is a minimal Gröbner basis of , then .
First, let us briefly review how the case Zariski-closed over , works for . The ideal of vanishing polynomials on is denoted , and the goal is to determine the minimal basis of .
is in the minimal basis, where .
is in the monomial basis if and only if is not empty, and .
The proof in Da09 () builds explicitly a Gröbner basis from to prove these assertions. Let us carry through the case .
Let , Zariski-closed over . The ideal of vanishing polynomials on is denoted . The strategy can be summarized as follows:
to each couple of integers , associate “naturally” a subset that may be empty. If it is not, this determines a third integer and .
construct by interpolation a polynomial that vanishes on and that has for leading monomial.
prove that is equal to union the for each 3-uplets mentioned in 1.
As it stands in this paper, the constructed Gröbner basis is not reduced. Experimentally, it has however smaller coefficients (proved in the case of 2 variables in Da09 ()).
Let . It is clear that it defines a partition of and since is finite, that almost all are empty. Let be such that . The shortcut notations and will be convenient. Let and .
[Proof.] Given , there exists such that . Thus , whereas if , then . This shows that . Since , holds . Note that and depends on . For the next step, let us introduce for :
Again it is clear that is a partition of , and that almost all are empty. Thus, is a partition of and almost all are empty. Let be such that is not empty, and let while . Finally let and and define .
[Proof.] The first equality directly results from the fact that . As for the second, if then whereas must fulfill . The insight of the decomposition is that is in the monomial basis of . Let and be the sets:
Let us illustrate how the decomposition works on an oversimplified but eloquent enough example.
Consider the finite set of points described hereafter. Define , and the points , having integer coordinates in Figure 1.
The third coordinate of the points in is determined by the circled number inside the regions delimited by Figure 1. For example the point of coordinate is inside the region marked 3, hence determines a point . The point with coordinates is inside the region marked 2, hence the three coordinates are . The cardinality of is . The regions delimited in Figure 1 illustrates the decomposition defined at the beginning of § 2.2.
The decomposition above permits to set up Lagrange interpolation formulas. We recall the basics along with setting notations.
Given a finite set of points in , Zariski-closed over and of cardinal , the Lagrange basis on of the -dimensional -vector space is defined as:
An element is written in this basis , in particular:
With the notations of the § 2.1, given such that define:
The following Proposition contains the conclusion of Theorem (structure) of the introduction.
Defining , the leading monomial of is .
Moreover, define as the unique monic polynomial such that . Then, divides and .
[Proof.] The degree in of the polynomials are equal to for all and for all . Because , and hence , it follows that the leading monomial of the polynomial
is , for all .
Similarly, the degree in of all the polynomials is , yielding:
It follows that . By Equation (7), because , verifies . By definition of equal to the cardinal of , one gets .
Equation (8) clearly shows that divides . Let . Then and one sees that
The Chinese Remaindering Theorem implies that divides modulo , hence that .
The polynomial vanishes on .
[Proof.] First, the factor implies that vanishes on , since . Because , it remains to show that vanishes on .
2.4 Concluding proof
Let also be a minimal Gröbner basis of . Lemma 3 shows that . In this subsection, we show that is a minimal Gröbner basis of . It is sufficient to prove that , or with the notation of Definition 1, that . This is done by induction on .
Assume that . Then .
[Proof.] The proof for the general case is no more complicated than the case , it suffices essentially to replace by 3 in the proof of Lemma 12.
With the base case treated, the induction can be carried through to prove that:
In general, the equality also holds.
The proof occupies the remaining of the section. It goes by induction on . The previous lemma treats the base case . Assume that , and let . Define . Note that . Since , then is not empty, and since , neither is . Let and .
We summarize hereunder the next steps heading to the proof of Theorem 1.
The same equality holds for the -elimination ideals which denotes the intersection with .
By construction therefore the induction hypothesis supplies the equality: .
The set verifies by construction hence falls into the case of Lemma 4 which gives: .
Putting this in the equality 1. shows that: .
and using the equalities in 2. and yeilds the equality: . This is equivalent to the statement of Theorem 1.
In the strategy outlined in points 1.-6. above, only the two ones require a proof. The point 2. can be deduced from the point 1., therefore we focus on proving 1. in the following. This occupies the remaining of this section.
Let and . These ideals satisfy the equalities , and .
Consequently, letting and , also hold and .
[Proof.] All ideals are radical here and thanks to the separability assumptions (H), the Nullstellensatz is satisfied. It suffices thus to prove that and that .
To start with, the fact that is minimal in implies that and . In the course of lemma 3, it was shown that given , thereby . And moreover that for and , implying that . Thus, vanishes on . By construction, and are disjoint, thus vanishes on