An Algorithm to Calculate the Kernel of Certain Polynomial Ring Homomorphisms

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1 An Algorithm to Calculate the Kernel of Certain Polynomial Ring Homomorphisms Fausto Di Biase and Rüdiger Urbanke CONTENTS 1. Introduction 2. The Solution by Means of a Gröbner Basis over Kx;y 3. The Solution by Means of Gröbner Bases over Kx 4. Simulation Results and Discussion References We propose an improvement upon the standard algorithm for computing the kernel of a polynomial map, assuming that the map sends monomials into monomials. Rather than computing a Gröbner basis in the joint polynomial ring, and then selecting only the elements of interest, we show that a moderate number of iterations of the Buchberger algorithm in the variables of the domain suffices. This work was carried out while the authors were at Washington University, St. Louis, in the Mathematics and Electrical Engineering Departments, respectively. Di Biase thanks the C.N.R. (Italy) for support, through grants and INTRODUCTION We are interested in calculating a nite basis for the kernel of a ring homomorphism : K x! K y between polynomial rings K x := K[x 1 ; : : : ; x n ] and K y := K[y 1 ; : : : ; y m ]. It is well known [Adams and Loustaunau 1994; Conti and Traverso 1991] that this can be accomplished applying the Buchberger algorithm [Buchberger 1985] over the polynomial ring K x;y := K[x 1 ; : : : ; x n ; y 1 ; : : : ; y m ]. Unfortunately, the complexity of the Buchberger algorithm is a strongly increasing function of the number of variables. Hence, it would be useful to nd an algorithm operating on K x alone. The main result of this paper is that this can indeed be done in the special case in which the map is the extension of a semigroup homomorphism. We will see that in this case a moderate number (bounded by b 1 nc) of Buchberger algorithms over K 2 x is suf- cient to nd a basis for ker, and, hence, that for a large number of variables the proposed algorithm will be more ecient than the standard algorithm. For general information on Grobner bases, see [Buchberger 1985; Cox et al. 1991]. The proposed algorithm has many applications, e.g., in the area of integer programming [Conti c A K Peters, Ltd /96 $0.50 per page

2 228 Experimental Mathematics, Vol. 4 (1995), No. 3 and Traverso 1991; Natraj et al.; Thomas; Hosten and Sturmfels 1994] as well in the realm of sampling from conditional distributions [Diaconis and Sturmfels]. We start by shortly reviewing the standard solution in Section 2. In Section 3 we then present the new algorithm together with some examples. Finally, in Section 4 the running times of these two algorithms are compared and the results are discussed. 2. THE SOLUTION BY MEANS OF A GRÖBNER BASIS OVER K x;y Let f j = (x j ) 2 K y and j = x j f j 2 K x;y, for j = 1; : : : ; n. Denote by hf: : :gi the ideal generated by f: : :g. Lemma 2.1. Let G be a (reduced) Grobner basis for hf 1 ; : : : ; n gi in K x;y with respect to a term order that eliminates the y variables. Then G 0 = G \ K x is a (reduced) Grobner basis for ker. Proof. The Elimination Theorem [Cox et al. 1991, p. 114] implies that G 0 is a (reduced) Grobner basis for hf 1 ; : : : ; n gi \ K x. The claim then follows since hf 1 ; : : : ; n gi is equal to the kernel of the (unique) homomorphic extension e : K x;y! K y of, for which e (x j ) = f j and e (y i ) = y i. Example 2.2. Let : K[x 1 ; : : : ; x 4 ]! K[y 1 ; y 2 ] map x 1 ; : : : ; x 4 into y 3 1, y 2 1 y 2, y 1 y 2 2, and y 3 2, respectively, so that 1 = x 1 y 3 1, etc. Then G = f x x 2 x 4 ; x 2 x 3 + x 1 x 4 ; x x 1 x 3 ; x 4 y 3 2 ; x 4y 1 + x 3 y 2 ; x 3 y 1 + x 2 y 2 ; x 2 y 1 + x 1 y 2 ; x 3 y 1 y 2 2 ; x 2 y 2 1 y 2; x 1 y 3 1g is the reduced Grobner basis for hf 1 ; : : : ; 4 gi with respect to lex order, y 1 > y 2 > x 1 > > x 4. Hence, G 0 = f x x 2 x 4 ; x 2 x 3 + x 1 x 4 ; x x 1 x 3 g is the reduced Grobner basis for ker with respect to lex order, x 1 > > x 4. Example 2.3. Let : K[x 1 ; : : : ; x 6 ]! K[y 1 ; : : : ; y 4 ] map x 1 ; : : : ; x 6 to, respectively, y 2 1 y3 2 y5 3, y 5 1 y2 2 y3 3 y 4, y 4 1 y4 2 y2 3 y 4, y 1 y 2 3 y4 4, y 1 y 5 3 y3 4, and y 2 2 y2 4. The reduced Grobner basis G for hf 1 ; : : : ; 6 gi with respect to lex order y 1 > y 4 > x 1 > > x 6 consists of 1180 elements, and G 0 = fx 10 3 x6 4 x 9 2 x 5x 11 6 ; x14 1 x20 4 x 2 x 6 3 x19 5 x8 6 ; x 16 3 x18 5 x 14 1 x8 2 x14 4 x3 6 ; x26 3 x17 5 x 14 1 x17 2 x8 4 x14 6 ; x 36 3 x16 5 x 14 1 x26 2 x2 4 x25 6 ; x46 3 x4 4 x15 5 x 14 1 x35 2 x36 6 g is the reduced Grobner basis for ker with respect to lex order x 1 > > x 6. Note that the above procedure requires the calculation of a Grobner basis G of hf i gi in K x;y, but the actual solution G 0 is then just a small subset of G, namely G 0 = G \ K x. This is especially apparent in the second example, where jgj = 1180 and jg 0 j = THE SOLUTION BY MEANS OF GRÖBNER BASES OVER K x Here we present an alternative way to calculate ker when (x j ) is a monomial in K y for every j = 1; : : : ; n. The solution is gradually built up by a repeated application of the Buchberger algorithm over K x (as opposed to K x;y ). The eciency of the proposed algorithm is based on the (empirical) fact that the complexity of the Buchberger algorithm grows strongly in the number of variables, so that for a large number of variables it is more ecient to calculate a moderate number of Grobner bases over K x instead of one over K x;y. This is especially true for the memory requirements of the proposed algorithm (see Example 2.3). Let M x denote the set of all monomials in K x, and likewise M y. Let f j = (x j ) and assume that f j 2 M y, for j = 1; : : : ; n. Note that all the information about is contained in the m n matrix M, with nonnegative integer entries, given by the exponents of the monomials f 1 ; : : : ; f n. We use the usual compact multi-index notation for monomials: e.g., for = ( 1 ; : : : ; n ) 2 N n, the symbol x denotes the monomial x 1 1 : : : x n n. Denote by log the isomorphism between M x and

3 Di Biase and Urbanke: An Algorithm to Calculate the Kernel of Certain Polynomial Ring Homomorphisms 229 N n given by log x := for 2 N n. There is a similar map from M y to N m, which, by a slight abuse of notation, we will denote by log also. Let M 2 N mn be dened by M i;j = (log f j ) i, where M i;j is the entry of M in row i, column j. This matrix denes a Z-linear mapping : Z n! Z m, by the usual row-by-column multiplication: (u) := Mu: A Z-basis for ker can be calculated using a variant of an algorithm that calculates the Smith normal form of an integer basis [Cohen 1993, p. 72]. The connection between ker and ker is conveyed by a map ' : Z n! K x, which will be presently dened. For u 2 Z dene u + and u by u + = max(u; 0) and u = max( u; 0). These denitions extend naturally to elements in Z n if we apply them componentwise. Note that if u 2 Z n then u + ; u 2 N n, and that ( u) + = u and ( u) = u +. Now for any u 2 Z n we dene '(u) := x u+ x u : Any binomial (dierence of monomials) p = x x, where ; 2 N n, can be written (in a unique way) as p = x x = m p '(u p ), where m p 2 M x and u p 2 Z n. We will write p + for x, and p for x. Then clearly p + = x = m p x u p + and p = x = m p x u p. Let 0 denote the natural partial order on N n obtained by forming the product of n copies of N with its natural order. If ; 2 N n, we denote by _ the 0 -smallest element of N n such that 0 _ and 0 _. In symbols, ( _ ) j = maxf j ; j g for j = 1; : : : ; n: We set x _ x = x _. Lemma 3.1. Let p and q be binomials in K x. Then p _ q + p p + p _ q + q + q = m p;q '(u p + u q ) (3.1) for some m p;q 2 M x. Proof. Note that u p (u p +u q ) + = u q (u p +u q ). Therefore the left-hand side equals p _ q + p (p + p ) + p _ q + where q + (q + q ) = p _ q + p + p _ q + p q + q = x u p+(u p +log(m p))_(u + q +log(m q)) x u q+(u +log(m p p))_(u + +log(m q q)) = x x (u p +u q ) + x (u p+u q ) = x '(u p + u q ); = up (up + uq) + +(u p +log(mp)) _(u + q +log(mq)) = uq (up + uq) +(u p +log(mp)) _(u + q +log(mq)): To see that 2 N n note that (u p + log(m p )) _ (u + q +log(m q )) is increasing (with respect to 0 ) in m p and m q. Hence, it suces to show that 2 N n in the case m p = m q = 1. Using the facts that u p _ u + q = u p + (u + q u p ) + and (u p + u q ) + = (u + p u q ) + + (u + q u p ) +, we get that = u + p (u + p u q ) + and, hence, 2 N n. Note that the S-polynomial [Cox et al. 1991, p. 82] of p and q, as well as the reduction of p with respect to q (if possible), can be written in the form of equation (3.1). This shows that if we calculate the Grobner basis of a set of elements of the form m i '(u i ), where m i 2 M x and u i 2 Z n, each element of this Grobner basis will also have this form, since this calculation can be done by computing a sequence of S-polynomials (possibly) followed by reductions. This can be seen in Examples 2.2 and 2.3 (each j is a binomial in K x;y ). We also have the following special case. (The support of 2 N n, denoted by supp, is the set of indices for which the corresponding component of is not zero.) Corollary 3.2. Let p = '(u p ); q = '(u q ) for some u p ; u q 2 Z n. If supp u + p and supp u q are disjoint (or supp u p is disjoint from supp u+ q ) then '(u p + u q ) 2 h'(fu p ; u q g)i: Proof. From Lemma 3.1 with m p = m q = 1 we have (using the same notation as in the previous proof) = u + p (u + p u q ) +. The hypothesis on

4 230 Experimental Mathematics, Vol. 4 (1995), No. 3 the supports implies that (u + p u q ) + = u + p and, hence, = 0. Example 3.3. Let u = (1; 2; 3; 1) = u + u = (1; 0; 3; 0) (0; 2; 0; 1) and v = (2; 1; 0; 2) = v + v = (2; 1; 0; 0) (0; 0; 0; 2). Since supp u + \ supp v = f1; 3g \ f4g =?; it follows from Corollary 3.2 that '(u + v) = '((3; 1; 3; 3)) = x 3 1 x3 3 x 2 x 3 4 belongs to hfx 1 x 3 3 x 2 2 x 4; x 2 1 x 2 x 2 4gi. Indeed, this binomial equals x 2 1(x 1 x 3 3 x 2 2 x 4) + x 2 x 4 (x 2 1 x 2 x 2 4). In the next theorem we see that the connection between ker and ker rests on the fact that ker is the ideal generated by '(ker ). Theorem 3.4. ker = h'(ker )i: Proof. It is simply based on a telescopic identity. Details can be found in [Herzog 1970]. If K is a basis for ker consisting of k elements (to simplify notation, we will use K to denote a basis for ker as well as the matrix in Z kn whose rows are the vectors in K), then it is not true in general that '(K) will be a set of generators for h'(span K)i, as the next example shows. Example 3.5. Let be as in Example 2.2. Then log(f1 ) M = 1 log(f 2 ) 1 log(f 3 ) 1 log(f 4 ) 1 log(f 1 ) 2 log(f 2 ) 2 log(f 3 ) 2 log(f 4 ) = : Calculating a Z-basis for ker we get K = : Hence, '(K) = fx 1 x 4 x 2 x 3 ; x 2 2 x 1 x 3 g and, calculating the reduced Grobner basis with respect to lex order, x 1 > > x 4, we get the set G = fx 2 x 2 3 x 2 2 x 4; x 2 x 3 + x 1 x 4 ; x 2 2 x 1 x 3 g: Since this (reduced and therefore unique) Grobner basis does not equal the Grobner basis we calculated in Example 2.2, we conclude that h'(k)i is strictly contained in h'(span K)i. Example 3.6. There are many equivalent choices for K. More precisely, if A 2 Z kk and det A = 1, then K 0 = AK is an equivalent basis (span K 0 = span K), and we write K 0 K. Conversely, any two equivalent bases are related in this way. Hence, we may ask if there always exists a K 0 K such that h'(k 0 )i = h'(span K)i. Again the previous example shows that this is not true. But there is an important special case in which '(K) is already sucient to generate h'(span K)i. Theorem 3.7. Let K 2 N kn. Then h'(k)i = h'(span K)i: Proof. Let g i, i = 1; : : : ; k, denote the rows of K. It suces to prove that if '(v) 2 h'(k)i and i 2 f1; : : : ; kg then '(v g i ) 2 h'(k)i, since any u 2 span K can be achieved in this way by starting with v = 0. Hence, assume '(v) 2 h'(k)i. Note that g i 2 N n, so that supp g i \supp v + =?. Therefore, by Corollary 3.2, '(v + g i ) 2 h'(fv; g i g)i h'(k)i: To prove that '(v g i ) 2 h'(k)i, note that v g i = v + ( g i ), and that supp( g i ) + \ supp v = supp g i \ supp v =?: In general, given a K 2 Z kn there will not exist a K 0 K such that K 0 2 N kn. Nevertheless we can always nd an equivalent basis with all base vectors lying in the same orthant, as the next lemma shows. Lemma 3.8. Let K 2 Z kn. Then there exists a K 0 K such that each column of K 0 is either in N k or in ( N ) k. Proof. First note that, if the j-th column of K is the zero k-tuple, the j-th column of any K 0 K is an

5 Di Biase and Urbanke: An Algorithm to Calculate the Kernel of Certain Polynomial Ring Homomorphisms 231 element of N k (being also the zero vector). Hence, without loss of generality, we can assume that each column of K contains at least one nonzero entry. If we can show that there exists a K 0 K such that some row of K 0 has all nonzero entries then by adding suitable (positive) multiples of this row to all other rows (thereby not changing the row space) we can generate the desired equivalent matrix. To see that such a K 0 exists, consider the set fi : 1 i n and jk j;i j > 0g; called the support of the j-th row of K and denoted by supp g j. Dene K 0 by g 0 1 = g 1 and g 0 j = g j + a j 1 g 0 j 1 for j = 2; : : : ; k, where a j 1 is any integer such that supp g 0 j = supp g j [supp g 0 j 1 (for instance, a j 1 = 1 + max i jg j;i j). Clearly K 0 K and supp g 0 k = k[ j=1 supp g j = f1; : : : ; ng: Therefore all components of g 0 k are nonzero. Example 3.9. Let be as in Example 2.2 and Example 3.5. Then M = = is an equivalent basis for ker with both rows in the same orthant. Lemma Let K 2 Z kn, and assume there exists a nite set U span K such that h'(u)i = h'(span K)i. If G is the reduced Grobner basis for h'(u)i (with respect to some xed order <) then G = '( ~ U) for some ~ U span K. Proof. By the remark following Lemma 3.1, each element g in the reduced Grobner basis has the form g = m'(u) for some m 2 M x and u 2 span K. Assume that m 6= 1 for some g 2 G. Now, '(u) 2 h'(u)i implies that '(u) (and, hence, g) is reducible by G n fgg, contradicting the fact that G was already reduced. For j 2 f1; 2; : : : ; ng dene T j : Z n! Z n as the operator that switches the sign of the jth component of the vectors in Z n. Further, if p 2 K x has the form p = '(u) for some u 2 Z n, we dene T j (p) = '(T j u). Theorem Let K 2 Z kn and assume that there exists a nite set U span K such that h'(u)i = h'(span K)i. If G is the reduced Grobner basis for h'(u)i, with respect to a term order that eliminates x j, then ht j Gi = h'(span(t j K))i. Proof. First observe that, by Lemma 3.10, there is a nite subset U ~ span K such that G = '( U). ~ Thus T j G = '(T j U) ~ is well dened. Moreover, since T j U ~ span(tj K), we have ht j Gi h'(span(t j K))i: To prove the other inclusion, let u 2 span K. Since G is a Grobner basis, there exists a g 2 G that gives rise to a head reduction of '(u). Without loss of generality, we may assume that '(u) = x a j p q and g = x b jr s, where a; b 2 N, a b and p; q; r; s are monomials in K[x 1 ; : : : ; x j 1 ; x j+1 ; : : : ; x n ]. Thus p = rm for some m 2 K[x 1 ; : : : ; x j 1 ; x j+1 ; : : : ; x n ] and '(u) = x a j p q = (xb j r s)xa b j m + (x a b j sm q) = gx a b j m + '(~u) ~m; where ~m is a monomial in K[x 1 ; : : : ; x j 1 ; x j+1 ; : : : ; x n ]; ~u 2 span K by Lemma 3.1, and the leading term of '(~u) is strictly smaller (with respect to the chosen term order) than the leading term of '(u). We have '(T j u) = p x a j q and T jg = r x b js. Hence, '(T j u) = p x a j q = (r xb j = (T j g)m + '(T j ~u)x b j ~m: s)m + (sm xa b j q)x b j We see that '(T j u) 2 ht j Gi will follow from '(T j ~u) 2 ht j Gi: The latter follows by induction, from the observation that the same procedure can be applied to

6 232 Experimental Mathematics, Vol. 4 (1995), No. 3 '(~u) (since ~u 2 span K) and that the leading term of '(~u) is strictly smaller than the leading term of '(u). We are now ready to describe our algorithm to calculate ker. Let K be a basis for ker. By Lemma 3.8 there exists an equivalent basis K 0 such that each column of K 0 is either in N n or in ( N ) n. Let J f1; 2; : : : ; ng be the index set of all columns with negative entries, and let K 0 J be the matrix obtained from K 0 by reversing all signs in the columns indexed by J. By Theorem 3.7, h'(k 0 J)i = h'(span K 0 J)i: If J =? we are done. If J 6=?, let j be any element of J. Theorem 3.11 enables us to derive from '(KJ) 0 a nite set of generators for h'(span K 0 )i. Jnfjg Compute the Grobner basis for '(KJ) 0 with respect to a term order that eliminates x j and apply the operator T j to it. Proceeding recursively, we can calculate a nite set of generators for '(span KJ), 0 which by Theorem 3.4 equals ker. Summary of the Algorithm. 1. Calculate a basis K for ker. 2. Find an equivalent basis K 0 such that all rows of K 0 lie in the same orthant. 3. Let J be the index set of all columns with negative entries and let KJ 0 be the matrix obtained from K 0 by reversing the signs of the columns indexed by J. 4. Let G J = '(KJ) Until J =?, repeat this: Take j 2 J and let G Jnfjg be the result of T j operating on the reduced Grobner basis for G J with respect to a term order that eliminates x j ; then let J J n fjg. 6. Output G?, a generating set for ker. Example Let be as in Examples 2.2, 3.5, and 3.9. Then G f2g = '(K 0 f2g ) = '(T 2K 0 ) = fx 1 x 2 2 x 3 1; x 2 1 x3 2 x 4 1g. Calculating a Grobner basis of G f2g with respect to lex order, x 2 > x 1 > x 3 > x 4, we get f x 2 4 x 1+x 3 3 ; x 4 x 2 x 2 3 ; x 4x 1 x 2 +x 3 ; 1+x 1 x 2 2 x 3g: Hence, G? := T 2 G f2g = f x 2 4 x 1 + x 3 3 ; x 2x 4 x 2 3 ; x 4 x 1 + x 2 x 3 ; x x 1 x 3 g is a generating set for ker '. Calculating a reduced Grobner basis of G? with respect to lex order x 1 > > x 4, we obtain f x x 2 x 4 ; x 2 x 3 + x 1 x 4 ; x x 1 x 3 g, which agrees with our previous result derived in Example 2.2. Example Let be as in Example 2.3. Then Hence, M = B@ CA ; K = K 0 = G f2;5;6g = '(K 0 f2;5;6g) ; : = fx 9 2 x10 3 x6 4 x10 5 x11 6 1; x 14 1 x10 2 x4 3 x26 4 x20 5 x19 6 1g: Calculating a Grobner basis of G f2;5;6g with respect to lex order x 6 > x 1 > : : : > x 5 and then applying T 6 we get the following set G f2;5g : fx 61 2 x146 3 x x x ; x 23 2 x x 56 1 x62 4 x73; 5 x 98 1 x110 4 x x 38 2 x92x 3 6; x 42 1 x28 4 x x 15 2 x38 3 x2; 6 x 8 2 x x 14 1 x14 4 x18 5 x3 6 ; x28 1 x34 4 x x 7 2 x22 3 x5 6 ; x 14 1 x 2x 20 4 x x 6 3 x8 6 ; x9 2 x10 3 x6 4 x 5 + x 11 6 g: In the same way we can calculate G f2g, which has 22 elements, and nally G?, with 14 elements. Calculating a Grobner basis of G? with respect to lex order x 1 > > x 6 one can check that this result coincides with the one derived in Example SIMULATION RESULTS AND DISCUSSION The proposed algorithm requires the determination of at most b 1 nc Grobner bases over K 2 x. Based on the (empirical) fact that the complexity of the Buchberger algorithm is a strongly growing function of the number of variables, we conclude that it

7 Di Biase and Urbanke: An Algorithm to Calculate the Kernel of Certain Polynomial Ring Homomorphisms 233 is in general more ecient to evaluate b 1 nc Grobner bases over K x than one Grobner basis over 2 K x;y. In the special cases where either ker (hence also ker ) is trivial or ker is spanned by a single element in Z n, no determination of a Grobner basis is required in the proposed algorithm; more generally, we observe that the starting point of the proposed algorithm is a collection of binomials, whose cardinality is equal to the nullity of a generic linear map from Z n to Z m : namely, maxfn m; 0g. Table 1 compares the running times of the standard versus the proposed algorithm for some parameters of n and m. For each pair (n; m) the time listed is the median of the running times, in seconds, of 500 random examples, where each entry in the matrix M was chosen independently according to a uniform distribution on the set f0; : : : ; 5g. The simulation was performed on a NeXT computer (25 Mhz, 20 MBytes of RAM) using Mathematica [Wolfram 1991] (which uses lex order) to perform the Grobner basis calculations and PARI [Batut et al. 1993] to perform the calculation of the integer kernel. From this table we see that for all listed cases but the case (n; m) = (6; 2) the proposed algorithm performs better than the standard algorithm. As expected, the dierence in performance becomes the more signicant the larger m is compared to n. Beside the signicant decrease in running time, the proposed algorithm also requires a signicantly smaller amount of memory. For the mapping given in Example 2.3 the largest intermediate Grobner basis required for the proposed algorithm has 22 elements, compared to 1180 elements that the standard algorithm requires. The listed running times could be further reduced in several ways. Instead of using Mathematica to perform the Grobner basis calculation, one could use Macaulay [Bayer and Stillman], which is signicantly faster. More substantially, all occurring ideals are toric ideals, for which specialized Buchberger algorithms have been investigated [Conti and Traverso 1991; Hosten and Sturmfels 1994]. These speedups apply to the standard as (n; m) standard alg. present alg. (3; 2) (3; 3) (4; 2) (4; 3) (4; 4) (5; 2) (5; 3) (5; 4) (5; 5) (6; 2) (6; 3) (6; 4) (6; 5) (6; 6) TABLE 1. Comparison of running times between the standard and the present algorithms. Times listed are in seconds, and represent the median running times of 500 random examples where each entry of M is independent and uniformly distributed in the set f0; : : : ; 5g. well as to the proposed algorithm. There are at least two more potential ways in which the proposed algorithm can be made more ecient. First, it is known that the lexicographical ordering is in general not very ecient, so a potential improvement would be to replace lex order by a more ef- cient order which eliminates x j. Secondly, in the above simulations no special eort was made to choose a specic K 0 from the many possible equivalent bases in order to minimize the subsequent calculations. Although the original task of the proposed algorithm was to compute the kernel of a polynomial map, assuming that the map sends monomials into monomials, the algorithm also applies to ring homomorphisms : K x! K y;y 1, where K y;y 1 := K[y 1 ; : : : ; y m ; y 1 ; : : : ; 1 y 1 m ]: For these maps, the entries of the corresponding matrix M will be integers (rather than non negative integers), but the algorithm works as well.

8 234 Experimental Mathematics, Vol. 4 (1995), No. 3 Finally, we observe that in the standard method one could use the FGLM algorithm [Marinari et al. 1993], which given a Grobner basis with respect to a certain ordering produces a basis with respect to another ordering. Since hf 1 ; : : : ; n gi is already a Grobner basis (with respect to any term order that eliminates the x variables), one could apply this FGLM algorithm to hf 1 ; : : : ; n gi to compute a Grobner basis with respect to a term order that eliminates the y variables, rather than computing this basis from scratch. But it seems that no substantial improvement in running times results from this approach. Further, this approach is limited to the original case where the map sends monomials into monomials, and does not extend to the more general case. REFERENCES [Adams and Loustaunau 1994] W. W. Adams and P. Loustaunau, An Introduction to Grobner Bases, Graduate Studies in Math. 3, Amer. Math. Soc., Providence, [Batut et al. 1993] C. Batut, D. Bernardi, H. Cohen and M. Olivier, User's Guide to Pari-GP. This manual is part of the program distribution, available by anonymous ftp from the host megrez.ceremab. u-bordeaux.fr. [Bayer and Stillman] D. Bayer and M. Stillman, Macaulay: A Computer Algebra System for Algebraic Geometry. This manual is part of the program distribution, available by anonymous ftp from the host available by anonymous ftp from zariski.harvard.edu. [Buchberger 1985] B. Buchberger, \Grobner bases: An algorithmic method in polynomial ideal theory", Chapter 6 in Multidimensional Systems Theory (edited by N. K. Bose), D. Reidel, Dordrecht and Boston, [Cohen 1993] H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Math. 138, Springer, New York, [Conti and Traverso 1991] P. Conti and C. Traverso, \Grobner bases and integer programming", pp. 130{ 139 in Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, New Orleans, 1991 (edited by H.F. Mattson et al.), Lecture Notes in Computer Science 539, Springer, Berlin, [Cox et al. 1991] D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Math., Springer, New York, [Diaconis and Sturmfels] P. Diaconis and B. Sturmfels, \Algebraic algorithms for sampling from conditional distributions", to appear in Ann. Stat. [Herzog 1970] J. Herzog, \Generators and relations of abelian semigroups and semigroup rings," Manuscripta Math. 3 (1970), 175{193. [Hosten and Sturmfels 1994] S. Hosten and B. Sturmfels, \GRIN: An implementation of Grobner bases for integer programming", 1994 (for information contact Hosten at serkan@math.berkeley.edu). [Marinari et al. 1993] M. Marinari, H. Moller, and T. Mora, \Grobner bases of ideals dened by functionals with an application to ideals of projective points", Appl. Algebra Eng. Commun. Comput. 4 (1993), 103{145. [Natraj et al.] N. R. Natraj, S. R. Tayur and R. R. Thomas, \An algebraic geometry algorithm for scheduling in the presence of setups and correlated demands", to appear in Math. Programming. [Thomas] R. R. Thomas, \A geometric Buchberger algorithm for integer programming", to appear in Math. Oper. Res. [Wolfram 1991] S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed., Addison-Wesley, Reading, MA, Fausto Di Biase, Department of Mathematics, Princeton University, Princeton, NJ (biase@math.princeton.edu) Rudiger Urbanke, Room 2C-254, AT&T Bell Labs, 600 Mountain Avenue, P.O.Box 636, Murray Hill, NJ (ruediger@research.att.com) Received May 19, 1994; accepted in revised form July 21, 1995