Computing a Gr¨obner basis of a polynomial ideal over a principal domain Ihsen Yengui (1) January 19, 2005

Abstract The purpose of this paper is to give a simple decision procedure for the ideal membership problem for polynomial rings over principal domains. As a particular case, we solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker’s problem”, that is the problem of finding a decision procedure for the ideal membership problem for Z[X1 , . . . , Xn ]. The techniques utilized are easily generalizable to Dedekind domains.

Key words : Gr¨obner basis, ideal membership problem, principal domains, Dedekind domains.

Introduction This paper is a participation in the program of extending the class of rings over which a notion of generalized Gr¨obner basis can be computed. The concept of Gr¨obner basis was originally introduced by Buchberger in his Ph.D. thesis (1965) in order to solve the ideal membership problem for polynomial rings over a field [4]. The ideal membership problem has received considerable attention from the constructive algebra community resulting in algorithms that generalize the work of Buchberger. Our goal is to find a decision procedure for the ideal membership problem for polynomial rings over a principal domain. The case where the basic ring is Z is called “Kronecker’s problem” and has been treated by many authors [1, 2, 6, 7]. Let us outline the strategy of our method. Begin by noting that for a principal domain R with field of fractions F, a necessary condition so that f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ] is: f ∈ hf1 , . . . , fs i in F[X1 , . . . , Xn ]. Suppose that this condition is fulfilled, that is there exists d ∈ R \ {0} such that d f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ].

(0)

As in [1], since the basic ring is principal and a fortiori factorial, we can write d = upn1 1 · · · pn` ` , where the pi are distinct irreducible elements in R, u is invertible in R, and ni ∈ N. Another necessary conditions so that f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ] is: f ∈ hf1 , . . . , fs i in Rpi R [X1 , . . . , Xn ] for each 1 ≤ i ≤ `. Write: di f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ] for some di ∈ R \ pi R.

(i)

Since gcd(d, d1 , . . . , d` ) = 1, by combining equalities asserting (0), (1), . . . , (`) using a Bezout identity between d, d1 , . . . , d` , we can find an equality asserting that f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ]. Thus, the necessary conditions are sufficient and it suffices to treat the problem in case the basic ring is a discrete valuation domain. The method we give is an adaptation of the notions of Gr¨obner basis and S-polynomials, originally introduced by Buchberger, to discrete valuation domains. The undefined terminology is standard as in [5] and [8]. 1

Departement of Mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia // Email: [email protected]

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Computing a Gr¨obner basis over a principal domain

Gr¨ obner basis over a valuation domain

P Definitions 1. Let R be a ring, f = α aα X α a nonzero polynomial in R[X1 , . . . , Xn ], E a non empty subset of R[X1 , . . . , Xn ], and > a monomial order. (i) The X α (resp. the aα X α ) are called the monomials (resp. the terms) of f . (ii) The multidegree of f is mdeg(f ) := max{α ∈ Nn : aα 6= 0}. (iii) The leading coefficient of f is LC(f ) := amdeg(f ) ∈ R. (iv) The leading monomial of f is LM(f ) := X mdeg(f ) . (v) The leading term of f is LT(f ) := LC(f ) LM(f ). (vi) LT(E) := {LT(g), g ∈ E}. (vii) hLT(E)i := hLT(g), g ∈ Ei (ideal of R[X1 , . . . , Xn ]). Definitions 2. Let R be a valuation domain, f, g ∈ R[X1 , . . . , Xn ] \ {0}, I = hf1 , . . . , fs i a nonzero finitely generated ideal of R[X1 , . . . , Xn ], and > a monomial order. (i) If mdeg(f ) = α and mdeg(g) = β then let γ = (γ1 , . . . , γn ), where γi = max(αi , βi ) for each i. The S-polynomial of f and g is the combination: S(f, g) =

Xγ LM(f ) f

S(f, g) =

LC(g) X γ LC(f ) LM(f ) f

−

LC(f ) X γ LC(g) LM(g) g

if LC(g) divides LC(f ).

γ

X − LM(g) g if LC(f ) divides LC(g) and LC(g) does not divide LC(f ).

(ii) G = {f1 , . . . , fs } is said to be a Gr¨obner basis for I if hLT(I)i = hLT(f1 ), . . . , LT(fs )i. (iii) As in the classical division algorithm in F[X1 , . . . , Xn ] (F field) (see [5], page 61), for each polynomials h, h1 , . . . , hm ∈ R[X1 , . . . , Xn ], there exist q1 , . . . , qm , r ∈ R[X1 , . . . , Xn ] such that h = q1 h1 , + · · · + qm hm + r, where either r = 0 or r is a sum of terms none of which is divisible by any of LT(h1 ), . . . , LT(hm ). H The polynomial r is called a remainder of h on division by H = {h1 , . . . , hm } and denoted r = h . Lemma 1. Let R be a valuation domain and I = haα X α , α ∈ Ai an ideal of R[X1 , . . . , Xn ] generated by a collection of terms. Then a term bX β lies in I if and only if X β is divisible by X α and b is divisible by aα for some α ∈ A. Proof. It is obvious that the condition is sufficient. For proving the necessity, write bX β = Ps γi αi for some α , . . . , α ∈ A, c , a n 1P s i αi ∈ R \ {0}, and γi ∈ N . Necessarily, for each i=1 ci aαi X X 1 ≤ i ≤ s, γi + αi = β, and b = si=1 ci aαi . It is clear that for each 1 ≤ i ≤ s, X β is divisible by X αi . P Let v : R → (G, ≤) be the valuation corresponding to R. Since v(b) = v( si=1 ci aαi ) ≥ min1≤i≤s v(ci aαi ) = min1≤i≤s (v(ci ) + v(aαi )) ≥ min1≤i≤s v(aαi ), we infer that there exists 1 ≤ i ≤ s such that v(b) ≥ v(aαi ), and therefore b is divisible by aαi . ♦ Lemma 2 (Dickson’s Lemma for discrete valuation domains). Let R be a discrete valuation domain and I = haα X α , α ∈ Ai an ideal of R[X1 , . . . , Xn ] generated by a collection of terms. Then there exist α1 , . . . , αs ∈ A such that I = haα1 X α1 , . . . , aαs X αs i. Proof. The proof is by induction on n and is similar to that of the classical Dickson’s Lemma (see [5], page 69). Let sketch the proof for n = 1. Let v : R → N be the valuation corresponding to R and denote by β an element of A such that v(aβ ) = inf α∈A v(aα ). If {γ1 , . . . , γk } is the maximal subset of A such that γ1 < β, . . . , γk < β, then I = haγ1 X γ1 , . . . , aγk X γk , aβ X β i. Now assume that n > 1 and that the lemma is true for n − 1. Let J be the ideal in R[X1 , . . . , Xn−1 ] generated by terms aα X α (α ∈ Nn−1 ) for which aα X α Xnm ∈ I for some m ≥ 0, and so on like in the proof of Theorem 5 of [5] (page 69).

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Using Lemma 1 and Lemma 2, we generalize some classical results about the existence of Gr¨obner basis for ideals in polynomial rings over discrete valuations domains. Theorem 1. Let R be a discrete valuation domain , I a nonzero ideal of R[X1 , . . . , Xn ], and fix a monomial order >. Then, I has a Gr¨ obner basis G, and for each f ∈ R[X1 , . . . , Xn ], f ∈ I if and only if the remainder on division of f by G is zero. The following lemma will be of big utility since it is the missing key for the characterization of Gr¨obner bases by means of S-polynomials (see [5], page 82). Lemma 3. Let R be a valuation domain, > a monomial order, and f1 , . . . , fs ∈ R[X1 , . . . , Xn ] Ps such that mdeg(f ) = γ for each 1 ≤ i ≤ s. If mdeg( a i i i=1 fi ) < γ for some a1 , . . . , as ∈ R, then Ps a f is a linear combination with coefficients in R of the S-polynomials S(fi , fj ) for 1 ≤ i, j ≤ s. i=1 i i Furthermore, each S(fi , fj ) has multidegree < γ. Proof. Since R is a valuation domain, we can suppose that LC(fs )/LC(fs−1 )/ · · · /LC(f1 ). Thus for LC(fi ) i < j, S(fi , fj ) = fi − LC(f fj . j) Ps LC(f1 ) LC(f1 ) LC(f2 ) i=1 ai fi = a1 (f1 − LC(f2 ) f2 ) + (a2 + LC(f2 ) a1 )(f2 − LC(f3 ) f3 ) + · · · + (as−1 +

LC(fs−2 ) LC(fs−1 ) as−2

+ ··· +

LC(fs−1 ) LC(f1 ) LC(fs ) as−1 + · · · + LC(fs ) a1 )fs . LC(fs−1 ) LC(f1 ) LC(fs ) as−1 + · · · + LC(fs ) a1 = 0 since

+(as + But as +

LC(f1 ) LC(fs−1 ) a1 )(fs−1

−

LC(fs−1 ) LC(fs ) fs )

P mdeg( si=1 ai fi ) < γ. ♦

Theorem 2. Let R be a valuation domain , I = hg1 , . . . , gs i an ideal of R[X1 , . . . , Xn ], and fix a monomial order >. Then, G = {g1 , . . . , gs } is a Gr¨ obner basis for I if and only if for all pairs i 6= j, the remainder on division of S(gi , gj ) by G is zero. Buchberger’s Algorithm. Let R be a discrete valuation domain , I = hg1 , . . . , gs i a nonzero ideal of R[X1 , . . . , Xn ], and fix a monomial order >. Then, a Gr¨ obner basis for I can be computed in a finite number of steps by the following algorithm: Input: g1 , . . . , gs Output: a Gr¨obner basis G for hg1 , . . . , gs i with {g1 , . . . , gs } ⊆ G G := {g1 , . . . , gs } REPEAT G0 := G For each pair f 6= g in G0 DO G0

S := S(f, g) If S 6= 0 THEN G := G0 ∪ {S} UNTIL G = G0 Proof. It is exactly the same algorithm as in the case the basic ring is a field. The only modifications are in the definition of S-polynomials and in the divisions of terms. Just, note that this algorithm must terminate after a finite number of iterations due to Lemma 2. ♦ A natural question arising is : For a valuation domain R, is it always possible to compute a Gr¨ obner basis for each finitely generated nonzero ideal of R[X1 , . . . , Xn ] by Buchberger’s Algorithm in a finite number of steps ? In fact, for a discrete valuation domain, what makes Buchberger’s Algorithm work in a finite number of steps is the fact that the totally ordered group ( = Z) corresponding to this valuation is well-ordered (note that, conversely, a well-ordered group is isomorphic to Z). Unfortunately, if the

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Computing a Gr¨obner basis over a principal domain

totally ordered group corresponding to the valuation is not archimedian, Buchberger’s Algorithm does not always work in a finite number of steps as can be seen by the following example. Example 1. Let V be a valuation domain with a corresponding valuation v and group G. Suppose that G is not archimedian, that is there exist a, b ∈ V such that: v(a) > 0, and ∀ n ∈ N∗ , v(b) > n v(a). Denote by I the ideal of V[X] generated by g1 = aX + 1 and g2 = b. Since S(g1 , g2 ) = ( ab )g1 − Xg2 = ab and ab is not divisible by b, then one must add g3 = ab when executing Buchberger’s Algorithm . In the same way, S(g1 , g3 ) = ( ab2 )g1 − Xg3 = ab2 and ab2 is not divisible by b nor by ab . Thus, one must add g4 = ab2 , and so on, we observe that Buchberger’s Algorithm does not terminate. Taking the particular case G = Z × Z equipped with the lexicographic order, a = (0, 1), and b = (1, 0). We can prove hLT(I)i is not finitely generated despite that I is finitely generated and that clearly hLC(I)i = hai (there is no such example in the literature). To check this, by way of contradiction, suppose that hLT(I)i = hh1 , . . . , hs i, hi ∈ I \ {0}, s ∈ N∗ . We can suppose that h1 , . . . , hs are terms, that is hi = LT(hi ) for each 1 ≤ i ≤ s. From Lemma 1, it follows that for each n ∈ N, there exists in ∈ {1, . . . , s} such that hin divides abn . We infer that there exists 1 ≤ i0 ≤ s such that hi0 is constant (hi0 ∈ V \ {0}) and such that ∀n ∈ N, hi0 divides

b . an

That is, v(hi0 ) ≤ (1, −n) ∀n ∈ N. It follows that there exists k ∈ N such that v(hi0 ) = (0, k) and hence there exists u invertible in V such that hi0 = uak . ½ k ½ k a ∈I a ∈I Now ⇒ ⇒ ak−1 ∈ I ⇒ · · · ⇒ a ∈ I ⇒ 1 ∈ I, a contradiction. aX + 1 ∈ I ak−1 (aX + 1) ∈ I ♦ As a consequence of this example, keeping the notations above, we know that a necessary condition so that Buchberger’s Algorithm terminates is that the group G is archimedian (this is in fact equivalent to dim V ≤ 1, see for example Proposition 8 page 116 in [3]). Moreover, we already know that a sufficient condition is that G is well-ordered (this is in fact equivalent to that V is a discrete valuation domain). This encourages us to set the following three conjectures : Conjecture 1. Let V be a valuation domain with corresponding valuation group G, n ∈ N∗ , and fix a monomial order > in V[X1 , . . . , Xn ]. Then the following assertions are equivalent: (i) It is always possible to compute a Gr¨ obner basis for each finitely generated nonzero ideal of V[X1 , . . . , Xn ] by the generalized version of Buchberger’s Algorithm for valuation domains in a finite number of steps. (ii) G is archimedian (⇔ dim V ≤ 1). Conjecture 2. Let V be a valuation domain (Pr¨ ufer domain) with a corresponding valuation group G, n ∈ N∗ , and fix a monomial order > in V[X1 , . . . , Xn ]. Then the following assertions are equivalent: (ii) dim V ≤ 1 (⇔ G is archimedian ). (iii) For each finitely generated ideal I of V[X1 , . . . , Xn ], the ideal {LT(f ), f ∈ I} of V[X1 , . . . , Xn ] is finitely generated. Conjecture 3. Let V be a valuation domain (Pr¨ ufer domain), n ∈ N∗ , and fix a monomial order > in V[X1 , . . . , Xn ]. Then for each finitely generated ideal I of V[X1 , . . . , Xn ], the ideal {LTC(f ), f ∈ I} of V is finitely generated.

I. Yengui

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The ideal membership problem and Gr¨ obner basis over a principal domain The ideal membership problem over a principal domain

As explained in the introduction, if R is a principal domain with field of fractions F, to answer the question Q: f ∈ ? hf1 , . . . , fs i in R[X1 , . . . , Xn ], one should first answer the question Q0 : f ∈ ? hf1 , . . . , fs i in F[X1 , . . . , Xn ]. If the answer to Q0 is negative then so is the answer to Q. If positive, there exists d ∈ R \ {0} such that d f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ]. (0) Since the basic ring is principal and a fortiori factorial, we can write d = pn1 1 · · · pn` ` , where the pi are distinct irreducible elements in R and ni ∈ N. The answer to the question Q is positive if and only if for all 1 ≤ i ≤ `, the answer to the question Qpi : f ∈ ? hf1 , . . . , fs i in Rpi R [X1 , . . . , Xn ], is positive. In case of positive answers, for each 1 ≤ i ≤ `, write: di f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ] for some di ∈ R \ pi R.

(i)

Since gcd(d, d1 , . . . , d` ) = 1, by combining equalities asserting (0), (1), . . . , (`) using a Bezout identity between d, d1 , . . . , d` , we can find an equality asserting that f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ]. As a conclusion, solving the ideal membership problem Q amounts to the resolution of a finite number of ideal membership problems Q0 ,Qp1 , . . . , Qp` over localizations of the basic ring R.

2.2

What is a Gr¨ obner basis over a principal domain ?

Let R be a principal domain with field of fractions F, and I = hf1 , . . . , fs i an ideal of R[X1 , . . . , Xn ]. We have seen that the first step to solve the ideal membership problem over R[X1 , . . . , Xn ] is to solve it over F[X1 , . . . , Xn ]. Let G0 = {g1 , . . . , gm } be a Gr¨obner basis for hf1 , . . . , fs i in F[X1 , . . . , Xn ]. For each 1 ≤ j ≤ m, h c write gj = cj and LC(gj ) = cj , where c, cj ∈ R \ {0} and hj ∈ R[X1 , . . . , Xn ]. Denoting by p1 , . . . , p` the distinct irreducible elements in R dividing one of c, c1 , . . . , cm , it is easy to see p1 , . . . , p` are the only irreducible elements in R that may appear as factors of the denominators of the quotients of the division of a polynomial in R[X1 , . . . , Xn ] by G0 . Let Gp1 , . . . , Gp` be Gr¨obner bases for hf1 , . . . , fs i respectively in Rpi R [X1 , . . . , Xn ], 1 ≤ i ≤ `, as explained in the first section. From the survey made previously, it is natural to suggest that the finite set G = {G0 , Gp1 , . . . , Gp` } will be called a Gr¨obner basis for I = hf1 , . . . , fs i in R[X1 , . . . , Xn ]. An element f ∈ R[X1 , . . . , Xn ] belongs to I if and only if all the remainders r0 , r1 , . . . , r` on division of f respectively by G0 , Gp1 , . . . , Gp` are zero (the set r = {r0 , r1 , . . . , r` } will be called the remainder on division of f by G).

2.3

An example

Example 2. This example illustrates the simplicity of our method. Let consider the ideal membership problem f = 5X 3 Y + 2X 2 + 3XY 2 + 4Y − 7 ∈? hf1 = 3XY + 4, f2 = 2X 2 + 3i in Z[X, Y ].

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Computing a Gr¨obner basis over a principal domain

Let fix the lexicographic order as monomial order with X > Y . By executing Buchberger’s Algorithm in Q[X, Y ], G0 = {f1 , f2 , 34 X − 32 Y, 98 Y 2 + 43 } is a Gr¨obner basis for hf1 , f2 i. The response to the ideal membership problem in Q[X, Y ] is positive. One obtains: 5 7 f = ( X 2 + Y )f1 − f2 . 3 3 By clearing the denominators, one gets: 3f = (5X 2 + 3Y )f1 − 7f2 .

(1)

It remains only to execute Buchberger’s Algorithm in Z(3) [X, Y ] as explained in this paper. One 2 obtains G3 = {f1 , f2 , 4X − 92 Y, 27 obner basis for hf1 , f2 i in Z(3) [X, Y ]. 8 Y + 4} as a Gr¨ 2 Thus, G = {{f1 , f2 , 34 X − 32 Y, 89 Y 2 + 43 }, {f1 , f2 , 4X − 92 Y, 27 obner basis for I = hf1 , f2 i 8 Y + 4}} is a Gr¨ in Z[X, Y ].

The response to the ideal membership problem in Z(3) [X, Y ] is positive. One obtains: 5 5 f = (Y − )f1 + ( XY + 1)f2 . 2 2 By clearing the denominators, one gets: 2f = (2Y − 5)f1 + (5XY + 2)f2 .

(2)

A Bezout identity between 2 and 3 is 3 − 2 = 1. Thus, (1) − (2) ⇒ f = (5X 2 + Y + 5)f1 + (−5XY − 9)f2 , a complete positive answer.

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The ideal membership problem and Gr¨ obner basis over a Dedekind domain

All what is made in this paper for principal domains can easily be generalized to Dedekind domains (see [8] for a constructive study of Dedekind domains). In order to avoid repetition, we keep the notations of the introduction, just suppose that R is a Dedekind domain. The factorization d = pn1 1 · · · pn` ` , is replaced by a decomposition of the principal ideal hdi into a finite product of nonzero prime ideals pi of R, say hdi =

` Y

pni i .

i=1

Of course, all the rings Rpi are discrete valuation domains in which the techniques of Section 1 apply. In case of positive answers in the rings Rpi , for each 1 ≤ i ≤ `, one can find di ∈ R \ pi such that di f ∈ hf1 , . . . , fs i in R[X1 , . . . , Xn ]. Since no prime of R contains the ideal hd, d1 , . . . , d` i, we infer that 1 ∈ hd, d1 , . . . , d` i. Moreover, since all the ideals of R are detachable (see [8], page 331), we can find explicitly an equality αd + α1 d1 + · · · + α` d` = 1, αi ∈ R, asserting that 1 ∈ hd, d1 , . . . , d` i, and so on exactly as in the principal domain case. For the notion of Gr¨obner basis for an ideal in polynomial ring over a discrete Dedekind domain, it is the same as in the principal domain case, just for an element a ∈ R \ {0}, replace the irreducible factors of a by the prime ideals of R appearing in the decomposition of the principal ideal hai into a finite product of nonzero prime ideals of R.

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References [1] M. Aschenbrenner, Ideal membership in polynomial rings over the integers, J. Amer. Math. Soc. 17 (2004), 407-441. [2] C. Ayoub, On constructing bases for ideals in polynomial rings over the integers, J. Number theory 17 (1983), no. 2, 204-225. [3] N. Bourbaki, “ Alg`ebre commutative”, Chapitres 5-6, Masson, Paris, 1985. [4] B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen polynomideal. Ph.D. thesis, University of Innsbruck, Austria, 1965. [5] D. Cox, J. Little and D. O’Shea, Ideals, varieties and algorithms, 2nd edition, New York, Springer-Verlag, 1997. [6] G. Gallo and B. Mishra, A solution to Kronecker’s problem, Appl. Algebra in Engrg. Comm. Comput. 5 (1994), no. 6, 343-370. [7] A. Kandry-Rody and D. Kapur, Computing a Gr¨obner basis of a polynomial ideal over a Euclidean domain, J. Symbolic Comput. 6 (1988), no. 1, 37-57. [8] R. Mines, F. Richman, W. Ruitenburg, A Course in Constructive Algebra, Universitext, Springer-Verlag, 1988.