Journal of Pure and Applied Algebra 164 (2001) 153–163www.elsevier.com/locate/jpaa

Degree reduction under specialization�

Elisabetta Fortunaa, Patrizia Giannia ;∗ Barry TragerbaDipartimento di Matematica, Universit�a di Pisa, Via Buonarroti 2, 56127 Pisa, ItalybT.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA

Abstract

We examine the degree relationship between the elements of an ideal I ⊆R[x] and the elementsof ’(I) where ’ :R → R is a ring homomorphism. When R is a multivariate polynomial ringover a 3eld, we use this relationship to show that the image of a Gr4obner basis remains aGr4obner basis if we specialize all the variables but one, with no requirement on the dimensionof I . As a corollary we obtain the GCD for a collection of parametric univariate polynomials.We also apply this result to solve parametric systems of polynomial equations and to reexaminethe extension theorem for such systems. c© 2001 Elsevier Science B.V. All rights reserved.

MSC: 13P10; 68W30

0. Introduction

Many authors have investigated properties of Gr4obner bases under specialization[1–3,5–11]. Most of them have focused on su<cient conditions to guarantee that theleading term ideal commutes with the specialization, i.e. ’(Lt(I)) = Lt(’(I)), sincein that case the image of a Gr4obner basis is still a Gr4obner basis for the specializedideal. In this paper, we investigate what information remains in a Gr4obner basis underspecialization even if the leading term ideal is not preserved. In particular, we willshow that, when we specialize all the variables but one, the image of a Gr4obner basisis always a Gr4obner basis. In the case of zero-dimensional ideals, results of this typewere obtained in Refs. [8,5], and later extended in Ref. [3].

� This work was partially supported by MURST.∗ Corresponding author.

E-mail addresses: fortuna@dm.unipi.it (E. Fortuna), gianni@dm.unipi.it (P. Gianni), bmt@watson.ibm.com(B. Trager).,

0022-4049/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S0022 -4049(00)00151 -1

154 E. Fortuna et al. / Journal of Pure and Applied Algebra 164 (2001) 153–163

Since we are concerned with situations where Lt(I) changes, we need to examinehow the degrees of polynomials behave under specialization. We will see that theminimum degree drop is an invariant of the ideal and that every polynomial in theimage ideal has an inverse image that drops in degree by the same amount.This concept of degree drop is then applied to Gr4obner bases under specialization. If

we specialize all the variables but one, we show that the specialized ideal is generatedby any non-zero polynomial of least degree in the image of the Gr4obner basis and thuswe obtain the GCD of a set of parametric univariate polynomials. We then examinesystems of polynomial equations with parametric coe<cients. We show that an extendedversion of the classical back substitution process can be nearly always used to solvesuch systems. The problematic situations arise when the block containing the variableto be solved for vanishes; we give some speci3c examples exhibiting this behavior.

1. Specializations and degree reduction

Let R be an integral domain and ’ :R → R a ring homomorphism; we will denote by’ also its natural extension to the polynomial ring R[x] obtained by de3ning ’(x)= x.For any f ∈ R[x] such that ’(f) �= 0 we will denote by �(f) the integer indicatinghow much the degree of f decreases under specialization, i.e.

�(f) = degf − deg’(f):We are interested in seeing how this gap changes when f varies in some ideal I ⊆R[x].At 3rst we will show that, if there exists g ∈ I with �(g)¡�(f), then there exists anew polynomial f̃ ∈ I such that �(f̃)¡�(f) and ’(f̃) = c’(f) with c ∈ R− {0}.It will be useful to write the polynomials f and g in a way especially convenient

for the next computations.Let s= deg’(g), so that �(g) = deg g− s. Let us write g in the form

g= G0 + xsG1;

where degG0¡s; consequently deg g= s+ degG1.If we denote m= degf − �(g) (where m ≥ 0 since �(g)¡�(f)), we can write f

in the form

f = F0 + xmF1;

where degF0¡m and hence degf = m+ degF1.Observe that(i) degG1 = deg g− s= �(g) = degf − m= degF1;(ii) ’(F1) = 0 because otherwise deg’(f) ≥ m = degf − �(g)¿ degf − �(f) =

deg’(f);(iii) ’(G1) ∈ R − {0} in fact ’(G1) �= 0 because otherwise ’(g) = ’(G0); moreover

deg’(G1)¿ 0 would imply deg’(g)¿s.

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Using these notations and preliminary remarks, let us 3rst show that, if �(g)¡�(f),we can 3nd f̃ with �(f̃)¡�(f) and ’(f̃)= c’(f); c ∈ R−{0}, when the degree ofg does not exceed the degree of f:

Lemma 1.1. Let f; g ∈ R[x] and assume that �(g)¡�(f). If deg g ≤ degf; thenthere exists f̃ ∈ (f; g) such that(1) deg f̃¡ degf;(2) ’(f̃) = c’(f) for a suitable c ∈ ’(R)− {0};(3) �(f̃)¡�(f).

Proof. Denote r = degf − deg g ≥ 0 and let f̃ = fG1 − gF1xr: Since m = r + s, itimmediately follows that f̃ = (F0 + xmF1)G1 − (G0 + xsG1)F1xr = F0G1 − G0F1xr .Using (i), we observe that

deg(F0G1) = deg(F0F1)¡m+ degF1 = degf

and also

deg(G0F1xr) = deg(G0G1xr)¡ deg g+ r = degf:

Therefore, deg f̃¡ degf. Moreover, using (ii), we have

’(f̃) = ’(F0G1)− ’(G0F1xr) = ’(F0)’(G1) = ’(G1)’(f)

which, by (iii), proves (2). In particular �(f̃)¡�(f).

If deg g¿ degf, the previous argument cannot be used, unless we 3rst modify gand produce a new g′ still verifying the condition �(g′)¡�(f) and such that deg g′ ≤degf. This will be possible by a repeated use of

Lemma 1.2. Let f; g ∈ R[x] and assume that �(g)¡�(f). If deg g¿ degf; thenthere exists g̃ ∈ (f; g) such that(1) deg g̃¡ deg g,(2) �(g̃)¡�(f).

Proof. Denote p= deg g− degf¿ 0 and let

g̃= fG1xp − gF1 = F0G1xp − G0F1;

where the second equality is due to the fact that m+p= s. Arguing as above, we have

deg(F0G1xp) = deg(F0F1) + p¡ degf + p= deg g

and also

deg(G0F1) = deg(G0G1)¡ deg g

so that deg g̃¡ deg g. Moreover,

’(g̃) = ’(F0G1xp)− ’(G0F1) = ’(G1)xp’(F0) = ’(G1)xp’(f):

Hence, �(g̃) = deg g̃ − deg’(g̃)¡ deg g − deg’(f) − p = deg g − degf +�(f)− p= �(f).

156 E. Fortuna et al. / Journal of Pure and Applied Algebra 164 (2001) 153–163

By a combined use of Lemmas 1.1 and 1.2 we can get the following result, whichproves the existence of a polynomial f̃ such that ’(f̃) = c’(f) and satisfying notonly the condition �(f̃)¡�(f) but the stronger condition �(f̃) ≤ �(g).

Theorem 1.3. Let f; g ∈ R[x] and assume that �(g)¡�(f). Then there exists f̃ ∈(f; g) such that(1) �(f̃) ≤ �(g),(2) ’(f̃) = c’(f) for a suitable c ∈ ’(R)− {0}.

Proof. First of all, let us construct a polynomial f̃ ∈ (f; g) such that ’(f̃) = c’(f)with c �= 0 and �(f̃)¡�(f).If deg g ≤ degf, we get f̃ by directly applying Lemma 1.1 to f and g.If deg g¿ degf, we can repeatedly apply Lemma 1.2 until we get g′ ∈ (f; g) such

that deg g′ ≤ degf and �(g′)¡�(f). This allows to proceed as before, that is to applyLemma 1.1 to the polynomials f and g′ and so obtain f̃ such that ’(f̃)= c’(f) and�(f̃)¡�(f).Now compare �(f̃) with �(g).If �(f̃) ≤ �(g), we are done.If �(f̃)¿�(g), the polynomials f̃ and g verify the hypothesis of the theorem, so

we can start again with these polynomials, remarking that �(f̃) is strictly lower thanthe original �(f), so that we obtain the thesis after a 3nite number of steps.

De nition 1.4. For any ideal I ⊆R[x], let �I =min{�(h) | h ∈ I; ’(h) �= 0}.

So �I is an invariant we can associate to any ideal I ⊆R[x] and which tells us that,under the specialization ’, the degree of every polynomial in I decreases by at least�I . With reference to the natural question whether any polynomial p ∈ ’(I) is thespecialization of a suitable polynomial in I whose degree drops by exactly �I , fromTheorem 1.3 we get:

Corollary 1.5. Let I be an ideal of R[x] and ’ :R → R a ring homomorphism.(1) For any p ∈ ’(I) there exist q ∈ I and c ∈ ’(R)−{0} such that ’(q)= cp and

�(q) = �I .(2) Let K be a @eld contained in R and assume that ’(R)⊆K and ’|K = id. Then

for any p ∈ ’(I) there exists h ∈ I such that ’(h) = p and �(h) = �I .

Proof. Let g ∈ I be such that �(g) = �I and let f ∈ I be such that ’(f) = p. If�(f) = �I , we are done taking q = f. If on the contrary �(f)¿�I , Theorem 1.3immediately implies that ’(q) = cp, with q and c as requested in (1). As for (2), theadditional hypothesis assures that c ∈ K−{0}; so it is su<cient to take h= c−1q.

In particular, in the case when ’(R)⊆K, if there exists g ∈ I whose leading co-e<cient does not vanish when specialized (i.e. �(g) = 0), then any p ∈ ’(I) is thespecialization of a suitable f ∈ I such that degf = deg’(f).

E. Fortuna et al. / Journal of Pure and Applied Algebra 164 (2001) 153–163 157

Remark 1.6. In the case when �I =0 the result of Corollary 1.5 can also be obtainedusing subresultant theory. But this theory gives no information in the case when theleading coe<cients of both polynomials vanish under the specialization. Thus, Theorem1.3 which also handles this situation can be viewed as a generalization of the propertiesof specializations deducible from the theory of subresultants.

2. Gr*obner bases under specializations

In this section we will apply the results obtained in Section 1 to the problem of3nding conditions for the specialization of a Gr4obner basis to be a Gr4obner basis. Wewill focus our attention on particular specializations, i.e. when all the variables but thehighest one are specialized and we will prove that no condition on the dimension ofthe ideal is necessary.Let I be an ideal in the polynomial ring K[x; T ], where K is a 3eld and T=(t1; : : : ; tr).

For any � ∈ Kr , denote by ’� the specialization that evaluates the variables T at �;precisely, ’� is de3ned by

’�(f(x; T )) = f(x; �) ∀f ∈ K[x; T ]:

Let ¿ be a block order with respect to x and T with x¿T , i.e.

xaTB¿xcTD ⇔ a¿c or (a= c and TB¿2 TD);

where ¿2 is an arbitrary term order on the T -variables.We will denote by lt(f) the leading term of f with respect to ¿, by ltx(f) and by

degx f, respectively, the leading term and the degree of f with respect to x and byLt(I) the ideal generated by {lt(f) | f ∈ I}.Let us show that, for any � ∈ Kr , the specialization of a Gr4obner basis of I is

always a Gr4obner basis of ’�(I), with no condition either on the specialization or onthe ideal, in particular on its dimension.

Theorem 2.1. Let I be an ideal in K[x; T ] and let G={g1; : : : ; gs} be a GrBobner basisof I with respect to the @xed block order ¿. Let � ∈ Kr and assume that ’�(I) �= 0;denote by gm the smallest (w.r.t. ¿) polynomial in G such that ’�(gm) �= 0. Then’�(gm) generates ’�(I) and �I = �(gm).

In particular ’�(G) is a GrBobner basis of ’�(I).

Proof. Let p be a generator of the principal ideal ’�(I). By Corollary 1.5 applied toI ⊆K[T ][x], there exists q ∈ I such that ’�(q) = p and �(q) = �I .As G is a Gr4obner basis, necessarily there exists i ∈ {1; : : : ; s} such that degx gi ≤

degx q and ’�(gi) �= 0. In particular, the polynomial gm is such that degx gm ≤ degx q.Moreover deg’�(gm) ≥ deg’�(q), since ’�(q) generates ’�(I). So

�I ≤ �(gm) = degxgm − deg’�(gm) ≤ degxq− deg’�(q) = �(q) = �I ;

which implies that deg’�(gm)=deg’�(q) and hence that ’�(gm) generates ’�(I).

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Theorem 2.1 shows that, for any specialization ’�, the ideal ’�(I) is generated byone of the polynomials in ’�(G) which is the specialization of a polynomial in Gwhose leading coe<cient (w.r.t. x) may vanish under specialization and whose degreereduction under ’� is measured by the invariant �I . Consequently ’�(I) is generatedby any non-zero polynomial in ’�(G) of least degree.Observe that

Proposition 2.2. If Ltx(I) denotes the ideal generated by the leading terms of thepolynomials in I with respect to x; then

�I = 0 ⇔ Lt(’�(I)) = ’�(Ltx(I)):

Proof. If �I = 0, from Theorem 2.1 it follows that degx gm = deg’�(gm). So we have

Lt(’�(I)) = (lt(’�(gm))) = (’�(ltx(gm)))⊆’�(Ltx(I))⊆Lt(’�(I)):

The converse is trivial.

The condition Lt(’�(I))=’�(Ltx(I)) was used by Gianni [5] and Kalkbrener [8] inorder to prove that ’�(G) is a Gr4obner basis when I is 0-dimensional. For example,in the proof by Gianni the hypothesis of 0-dimensionality is used to show that thereexists a polynomial g ∈ I such that ’�(g) generates ’�(I) and degx g = deg’�(g);consequently, Lt(’�(I)) = ’�(Ltx(I)), which is a su<cient test-condition for ’�(G)to be a Gr4obner basis. So Theorem 2.1 extends the result by Gianni and Kalkbrener,independently of the value of �I .

3. Applications

In this section, we want to apply the results obtained so far to some “parametric”problems, more precisely to the computation of the “parametric” GCD of a set ofpolynomials and to the resolution of a system of polynomial equations with parametriccoe<cients.A 3rst consequence of our results is a characterization of the parametric GCD of a

set of polynomials:

Proposition 3.1 (Parametric GCD). Let f1; : : : ; fk be a set of polynomials in K[x; T ].Let G = {g1; : : : ; gs} be a GrBobner basis of I = (f1; : : : ; fk) with respect to a blockorder ¿. Let � ∈ Kr and assume that ’�(I) �= 0. If gm is the smallest (w.r.t. ¿)polynomial in G such that ’�(gm) �= 0; then

GCD(f1(x; �); : : : ; fk(x; �)) = gm(x; �):

The last result can also be rephrased in terms of comprehensive Gr4obner bases [13]:

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Corollary 3.2. A GrBobner basis of an ideal I in K[x; T ] with respect to a block orderfor x; T is a comprehensive GrBobner basis in K[T ][x]; i.e. considering the T ’s asparameters.

As a second consequence of our results, let us show that it is possible to weakenthe hypothesis of the classical extension theorem (see for instance [4]) preserving itsconclusions, if a Gr4obner basis of I is known. In fact we have:

Theorem 3.3 (Gr4obner Extension Theorem). Let I be an ideal in K[x; T ] and let G={g1; : : : ; gs} be a GrBobner basis w.r.t. a block order ¿. Let I1 = I ∩ K[T ] be the@rst elimination ideal. Let � ∈ V (I1) and assume that ’�(I) �= 0. Denote by gm thesmallest (w.r.t. ¿) polynomial in G such that ’�(gm) �= 0. Then; if a ∈ K; we havethat

(a; �) ∈ V (I) ⇔ ’�(gm)(a) = 0:

Proof. It is an immediate consequence of Theorem 2.1 since ’�(gm) generates ’�(I).

Corollary 3.4 (Extension Theorem). Let K be an algebraically closed @eld; I an idealin K[x; T ] and I1 = I ∩ K[T ] the @rst elimination ideal. Let � ∈ V (I1) and assumethat �I = 0. Then there exists a ∈ K such that (a; �) ∈ V (I).

Proof. Let G be a Gr4obner basis of I w.r.t. a block order for x; T . By Theorem2.1 there exists a polynomial gm ∈ G such that deg’�(gm) = degx gm. Observe thatdegx gm �= 0, because otherwise V (I1) = ∅. So the thesis follows immediately fromTheorem 3.3.

Finally, we apply our results to the problem of the resolution of parametric systemsof polynomial equations.Let X =(x1; : : : ; xn); T =(t1; : : : ; tr) and f1; : : : ; fk ∈ K[X; T ]. Consider the system of

parametric polynomial equations "={f1=0; : : : ; fk=0}. Assume G is a Gr4obner basisfor the ideal I = (f1; : : : ; fk) with respect to an order that is a block order for X; Twith the X -variables lexicographically ordered with x1¿ · · ·¿xn. Then it is known(see [12]) that the elements of G can be collected in blocks G = {B1; : : : ; Bn}, whereBi = {gi;1(xi; : : : ; T ); : : : ; gi; ki(xi; : : : ; T )}⊂K[xi; : : : ; xn; T ], in such a way that the idealgenerated by

⋃j¿i Bj is the ith elimination ideal Ii = I ∩ K[xi+1; : : : ; xn]. Every time

we specialize the parameters to � ∈ Kr , let us call ith partial solution any element%(i) ∈ V (Ii)⊆Kn−i, i.e. any vector %(i) such that gj;m(%(i); �)=0 for all gj;m ∈ Bj forj¿ i.In order to solve the given system, we can think of using the usual strategy of back

substitution, that is extending any ith partial solution to all the possible (i−1)th partialsolutions and so forth. Notice that after specializing the parameters it is no longertrue that the ideal generated by

⋃j¿i ’�(Bj) is necessarily the ith elimination ideal of

160 E. Fortuna et al. / Journal of Pure and Applied Algebra 164 (2001) 153–163

’�(I). However, our results assure it is possible to decide if such an extension stepcan be done (and in this case to 3nd all the possible extensions) or excluded, unlessthe next block Bi vanishes under the specialization ’%(i); �.

Theorem 3.5. Let " = {f1 = 0; : : : ; fk = 0} be a system of parametric polynomialequations; with fi ∈ K[X; T ]; X = (x1; : : : ; xn) and T = (t1; : : : ; tr). If we specialize theparameters to � ∈ Kr ; let %(i) be a ith partial solution and ’%(i); � be the correspondingspecialization homomorphism. Suppose that ’%(i); �(Bi) �= 0 and let h be the smallest(w.r.t. ¿) polynomial in Bi such that q= ’%(i); �(h) �= 0. Then(1) if deg q= 0; the ith partial solution cannot be extended;(2) if deg q¿ 0; the ith partial solution can be extended to an (i − 1)th partial

solution using any zero of q(xi).

So, as long as the ith partial solution does not annihilate the block Bi, the previoustheorem shows how to proceed. Problems can arise when this condition is not true,because this forces us to go back until we 3nd a non-zero block, but then we are nolonger in the situation examined in Section 2 of specializing all the variables but one.

Example 3.6. Let f1 = ax2 + x + y; f2 = bx + y and I = (f1; f2)⊆K[x; y; a; b]. TheGr4obner basis for I w.r.t. the lex order consists of the blocks

B1 = {ax2 + x + y; axy − by + y; bx + y}; B2 = {ay2 + b2y − by}:

If we choose a=0; b=0 (i.e. �=(0; 0)), then ’�(B2)={0}, so the evaluated block doesnot give any relation for y. Nevertheless, notice that in this case ’�(B1)={x+y; y; y};so in it we 3nd a relation for y.

Let us conclude the section by showing that, for parametric polynomial systemsin 2 variables, provided we know that ’�(I) is zero-dimensional, it is possible togive a procedure to compute the solutions even when the block B2 vanishes underspecialization. In that case, as in Example 3.6, we can get a relation for y from theblock B1, in such a way that all the possible partial solutions of ’�(I) can be foundamong the roots of the relation so found:

Proposition 3.7. Let I be an ideal in K[x; y; T ] and G = (g1; : : : ; gs; h1; : : : ; ht) be aGrBobner basis for I w.r.t. a block order ¿; with (h1; : : : ; ht) = I ∩ K[y; T ]. For any� ∈ Kr ; denote by ’� : K[y; T ] → K[y; T ] the homomorphism de@ned by ’�(y) = yand ’�(T )=�. Assume that ’�(hi)=0;∀i=1; : : : ; t. If ’�(I) is zero-dimensional; thenthere exists k ∈ {1; : : : ; s} such that ’�(gk) ∈ K[y]− {0}.

In particular if (%; �) ∈ V (I∩K[y; T ]) is a partial solution of the polynomial systemassociated with I; then % is a root of ’�(gk) ∈ K[y].

Proof. Since ’�(I) is zero-dimensional, there exists a non-zero polynomial p(y) ∈’�(I) ∩ K[y] of least degree. By Corollary 1.5, applied with R = K[y; T ], there exist

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polynomials q ∈ I and c ∈ K[y] such that ’�(q) = c(y)p(y) and �(q) = �I . So theconclusion follows from the next lemma.

We denote by K[Y ] any multivariate polynomial ring over the 3eld K.

Lemma 3.8. Let R = K[Y ] and let ’ :R → R be a ring homomorphism. Let I bean ideal of R[x] and denote by G = {g1; : : : ; gs} a GrBobner basis for I w.r.t. a blockorder. Assume that q=

∑si=1 bigi is a GrBobner representation of q ∈ I (in particular

degx (bigi) ≤ degx q whenever bigi �= 0). Let H = {i |’(bigi) �= 0}.(1) If ’(q) �= 0; then there exists i ∈ H such that �(gi) ≤ �(q).(2) If ’(q) ∈ R − {0} and �(q) = �I ; then ∀i ∈ H we have �(gi) = �I = degx gi (in

particular ’(gi) ∈ R).

Proof. Observe that H �= ∅ because ’(q) �= 0.(1) If �(gi)¿�(q) for all i ∈ H , then we would have

degx ’(q) = degx q− �(q)¿ degx q− �(gi) ≥ degx (bigi)− �(bigi) = degx ’(bigi)

which is impossible since ’(q) =∑

i∈H ’(bigi).(2) Since �(q) = �I , for all i ∈ H we have

�(bigi) ≥ �(q) = degxq ≥ degx(bigi) ≥ �(bigi);

therefore �(bigi) = �(q) = �I . Moreover �(bigi) ≥ �(gi), so that �(gi) = �I . Thendegx gi = degxq= �I , so that bi ∈ R and ’(gi) ∈ R.

From part (1) of Lemma 3.8 we immediately get

Corollary 3.9. Let R = K[Y ]. If I ⊆R[x] is an ideal; G = {g1; : : : ; gs} is a GrBobnerbasis for I w.r.t. a block order and ’ :R → R is a ring homomorphism; then �I =min{�(gi) |’(gi) �= 0}.

This result guarantees that the least degree drop �I for an ideal is attained by someelement of the Gr4obner basis, not only when specializing all the variables but one (seeTheorem 2.1), but also for any specialization. In particular �I is computable as theminimum of a 3nite set.

Remark 3.10. Proposition 3.7 can be used with Theorem 3.5 to increase the set ofproblems we can solve. Using the notation of Theorem 3.5, if ’%(i); �(Bi) = 0 but theideal generated by ’%(i); �(Bi−1) in K[xi−1; xi] is zero-dimensional, then Proposition 3.7applies and we 3nd a relation for xi in ’%(i); �(Bi−1).

Let us conclude by an example in which we cannot proceed as indicated in Remark3.10, even if the ideal obtained by specializing all the parameters is zero-dimensional:

Example 3.11. Let B1 = {ax2 + x+y; axz− by+ z; bx+ z}; B2 = {b2y+ az2− bz; cy+y− z}; B3 ={acz2 +az2 +b2z−bcz−bz}: These polynomials form a Gr4obner basis for

162 E. Fortuna et al. / Journal of Pure and Applied Algebra 164 (2001) 153–163

the ideal I=(B1; B2; B3)⊆K[x; y; z; a; b; c]. If we choose a=b=c=0 (i.e. �=(0; 0; 0)),the ideal ’�(I) is zero-dimensional. Observe that ’�(B3) = {0}, hence the evaluatedblock does not give any relation for z; moreover ’�(B2) = {y − z}, so that the idealgenerated by ’�(B2 ∪ B3) is not zero-dimensional. However, ’�(B1) = {x+ y; z; z}, soalso in this case a relation for z appears in an earlier block, even if, arguing as inthe proof of Proposition 3.7, we could only be sure that ’�(B1) contains at least onepolynomial in y and z, but not necessarily any polynomial only in z.

4. Conclusions

We have investigated which information remains in a Gr4obner basis under special-ization even if the leading term ideal is not necessarily preserved. When we specializeall the variables but one, the result is still a Gr4obner basis. When we work with para-metric systems of polynomial equations, we 3nd that, even if the specialized basis isno longer Gr4obner, often it can still be used to 3nd all the solutions. During the backsubstitution process, if each partial solution does not annihilate the next block, thenwe can proceed as if we had a Gr4obner basis, although in this case some partial solu-tions will not extend to complete solutions and will be discarded. The di<culty ariseswhen the block containing the next variable to be solved vanishes. This occurs whenthe specialization of one of the elimination ideals is not zero-dimensional even if thespecialization of the entire ideal does have a 3nite number of solutions. We have pre-sented a partial result which shows that under some conditions we can guarantee thatan equation for the missing variable will be found in the next block. The question ofhow much additional computation is required to solve the problem in general remainsan open question.

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