Cent. Eur. J. Math. •8(4) • 2010 • 780-785 DOI: 10.2478/s11533-010-0045-0

VERS ITA

Central European Journal of Mathematics

Border bases and kernels of homomorphisms and of derivations

Research Article

Janusz Zielinski1*

1 Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Torun, Poland

Received 29 December 2009; accepted 10 May 2010

Abstract: Border bases are an alternative to Grobner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.

MSC: 13N15, 13P10, 68W30, 12Y05, 12H05

Keywords: Border basis • Grobner basis • Factorizable derivation • Ideal of relations © Versita Sp. z o.o.

1. Introduction

Let k be a field. We will denote by k* the set k \ {0}. By k[X] we mean k[x1,..., xn], the polynomial ring In n variables, and by k(X) we mean k(x1,... ,xn), the field of rational functions. Let N designate the set of nonnegative integers. For a = (a1.....an) e Nn, we denote by Xa the monomial x^ ■■■xann G k[X].

Let R be a commutative k-algebra. A k-linear map d : R — R is said to be a k-derivation (or simply a derivation) of R if d(ab) = ad(b) + bd(a) for all a,b e R. By Rd we denote the kernel of the derivation d. It forms a ring and we call it the ring of constants of d. It is well known that k C Rd. A nontrivial constant of the k-derivation d is the element of the set Rd \ k. For any given derivation d : k[X] — k[X] there exists exactly one derivation d : k(X) — k(X) such that d\k[X] = d. By a rational constant of the derivation d : k[X] —> k[X] we mean the constant of its corresponding derivation d : k(X) - k(X).

In Section 2 we give the definition of a factorizable derivation. Important Lotka-Volterra derivations are examples of factorizable derivations. We show how to associate the factorizable derivation with any given derivation. The construction helps to establish new facts on our initial derivation, especially on its rational constants (see [7]). Furthermore we define

* E-mail: ubukrool@mat.uni.torun.pl

Springer

an Ideal of relations associated with a derivation and we estimate the number of generators of the Ideal of relations. Finally, we apply these considerations to the rational constants.

A border basis is a set of generators of an ideal in a polynomial ring, with some additional properties (more details in Section 3). The notion is somewhat analogous to the Grobner basis. There are some advantages of border bases over Grobner bases:

1. Grobner bases behave numerically unstable forming representation singularities. A small change in the coefficients of generators of an ideal leads to a big change in the reduced Grobner basis (see 6.4.1 of [6]). Whereas border bases may change continuously into one another when we continuously vary the coefficients of generators (6.4.22

of [6]).

2. An ideal which is symmetric with respect to swapping indeterminates, may not have any symmetric reduced Grobner basis, however has a symmetric border basis (6.4.2 and 6.4.22 of [6]).

3. Border bases are useful in statistics (design of experiments [5]). Order ideals appear in this field in a natural way and in simple shapes. Methods of border bases are very convenient to optimize many processes.

4. In some cases border bases algorithms are significantly faster than those of Grobner bases (see 19 and 20 of [4]). The reason lies in the fact that the border basis computation requires polynomials of much lower degrees (although there are often more polynomials). Both approaches were tested using the CoCoA system and in some instances one theory is more effective while, in some, the other.

5. When we compute all the border bases of an ideal, then as a consequence we also easily obtain all the reduced Grobner bases ([4]), whereas the reverse is not true.

The above examples indicate the merits of further investigation of border bases. In this paper we give some subsequent applications of border bases.

The main results of the paper, presented in Section 4, are Theorem 4.1 and the conclusions drawn from Propositions 4.1 and 4.2. They provide constructions of border bases in the following cases:

1. The restriction of an ideal to a polynomial ring in a smaller number of variables.

2. The intersection of two ideals.

3. The kernel of a homomorphism of polynomial rings (for example, an ideal of relations).

2. Factorizable derivations and ideals of relations

In this section k is a field of characteristic zero. A derivation d : k[X] — k[X] is called factorizable if d(xt) = xf, where fi G k[X] for i = 1,..., n. Now we show how to associate the factorizable derivation with a given derivation. Recall that for any field F and a derivation d : F — F, by a logarithmic derivative of the derivation d we mean the mapping L : F* — F defined by L(a) = d(a)/a for all a G F*. If y G k(X)* is of the form y = ax^1 .. .x^n, where a G k* and a1,... ,an G Z, then

L(y) = aiL(xi)+-----+ anL(xn).

Let d : k[X]^> k[X] be a derivation defined on the variables as follows:

d(xi) = mi1 + mi2 + • • • + miri

for i = 1,..., n, where mj for j = 1,..., ri, are monomials of the form ax^ . . . x^n, where a G k* and ... ,fin G N. Define the set

m^; 1 < i < n, 1 < j < r\ C k(X).

Let R be the smallest subring in k(X) containing k and S, that is, R = k[S]. Then d(xi)/xi e R for i = 1,..., n and the inclusion d(R) C R holds.

Now define the set S', obtained from the set S by normalization of the elements of S, that is, all coefficients are now equal to 1 (note that from all associated elements of S we obtain only one element of S') and then by crossing out, if it exists, the element from the field k. Obviously, R = k [S']. Let S' = {f1,... ,fp}. For i = 1,..., n we have

G ked + ky1 +-----h kfp,

where ed = 0 if no element from k* belongs to S and ed = 1 in the opposite case. The mapping d\R : R — R is well-defined and

d(fi) = ft(bted + al1f1 +-----h avfp)

for l = 1,..., p and some b1,..., bp, aij e k, where 1 < j < p.

We introduce new variables y1,..., yp. We will denote by k[Y] the polynomial ring k[y1,..., yp]. Let 5 : k[Y] — k[Y] be a derivation defined by

5(yi) = yi(bied + a^y1 +-----h a^yp)

for i = 1,..., p. With the above notations we call 5 the factorizable derivation associated with derivation d. For more details we refer the reader to [7].

Let d : k[X] — k[X] be a derivation and let S' = {f1,...,fp} be its corresponding set defined above. Let = xli1 .. .xf" = Xai for i = 1,..., p. Then we have the matrix a = (alj)1<i<p,1<j<n G Mpyn(Z).

Let R = k[f^,..., yp] C k(X). Define a homomorphism n : k[y-\,..., yp] — R by r/(yl) = ^ for i = 1,..., p. It is an epimorphism. Therefore, R X k[y-\,...,yp]/ker(r). By the ideal of relations associated with the derivation d we mean the kernel of the homomorphism r. We denote it by A(a). Hence, A(a) = jft e k[y^,..., yp];h(f1,..., fp) = 0 J . The ideal A(a) is prime and differential. It contains neither a linear form nor a nonzero monomial ([8]). By a monic binomial we mean a polynomial of the form Ya — Yb e k[Y], where a, b e Np.

Theorem 2.1 ([8], 5.7).

Let = X11,... ,fp = X1p e k (X), where a-],..., ap are pairwise distinct elements of the set Zn \ {(0,..., 0)}. If the ideal A(a) is nonzero, then it has (p — rankZa) algebraically independent a-homogeneous irreducible monic binomials.

It is shown in [8] that if A(a) = 0, then the derivation 5 has a nontrivial rational constant, and, an estimate can be given.

Theorem 2.2 ([8], 6.5).

If A[a] = 0, then the derivation 5 has (p — rankZa) rational constants which are algebraically independent over k.

The main application of factorizable derivations is determining whether the given derivations have nontrivial rational constants (in general, still an open problem). For instance, in [7], this tool gives a full and extensive description of all monomial derivations of k[x,y,z] with nontrivial rational constants, as well as other applications.

3. Border bases

A non-empty set of terms O in k[X] is called an order ideal if t e O implies t' e O for every term t' dividing t. This ideal is a monoid ideal, not a ring ideal. The border of O is the set of terms dO = (x^O U ... U xnO)\O. Let O be finite and dO = {b,..., br}. A set of polynomials G = {g^,..., gr} e k[X] is called an O-border prebasis if gi = bi + hi, where hi e k[X] satisfies Supp(hi) CO for 1 < i < r.

Let I be a zero-dimensional ideal of k[X] containing O-border prebasis G. We say that G is an O-border basis of I if the residue classes of the elements of O form a k-vector space basis of k[X]/I.

There are many equivalent conditions to the above definition;eighteen of them are given in [3]. We will use the condition of Proposition 3.1, called the Buchberger criterion for border bases. The S-polynomial of two distinct elements gi, gj G G is defined by

S(gi, gj) = (lcm(6j, bj)/bl)gl — (lcm(bi, bj)/bj)gj.

Two prebasis polynomials, gi and gj are neighbors if their border terms are related according to xpbi = xsbj or xpbi = bj for some indeterminates xp,xs.

Proposition 3.1 ([4], 4).

An O-border prebasis G = {g1,..., gr} is an O-border basis of an ideal I if and only if G C I and, for each pair of neighbors in G, there are constant coefficients ci G k such that S(gi, gj) = c1g1 + • • • + crgr.

For every zero-dimensional ideal I there exists an order ideal O supporting a border basis. An example is Oa{I} : = Tn \ LTa{I}, where, here and throughout, Tn denotes the monoid of terms and LTa{I} denotes the set of leading terms of I with respect to a term ordering a. If for a given O a border basis G of I exists, then G is uniquely determined and generates I.

In [4] there are several algorithms for computing border bases presented. We will employ the following Basis Transformation Algorithm ([4], Proposition 5).

Proposition 3.2.

Let I C k [X ] be a zero-dimensional ideal and let O = {t1,..., tm} be an order ideal. The following algorithm checks whether O supports a border basis of I and, in the affirmative, computes the O-border basis {g1,..., gr} of I.

(T1) Choose a term ordering a and compute Oa{I} := Tn \ LTa{I}.

(T2) If #(Oa{I}) = m, then return "O has the wrong cardinality to support a border basis of I," and stop.

(T3) Let Oa{I} = {s1,...,sm}. For 1 < p < m, compute the coefficients Tip G k of the normal form NFaI(tp) = TipSi. Let T be the matrix (Tip)-\<t,p<m.

(T4) If det T = 0, then return "O has the wrong form to support a border basis of I," and stop.

(T5) Let dO = {b1.....br}. For 1 < j < r, compute the coefficients fitj G k of NFaI(bj) = Pjsl. Let B be the

matrix (fiij)1<i<m1<j<r.

(T6) Compute (aj) = TB. Return gj := bj — Yi"i=1 aijti for 1 < j < r.

4. Operations on border bases

Let k[X, Y] = k[x1.....xn,y1.....ym]. If O is an order ideal in k[X, Y], then O n k[Y] is an order ideal in k[Y], If

I < k[X, Y] is a zero-dimensional ideal, then obviously the same is true for I n k[Y]. A term ordering a such that t >a t' for terms t G k[X], t' G k[Y] is, among many others, a = Lex. The following reduction requires only some set-theoretical condition on cardinality. Note also that we can just take O = Oa{I}.

Theorem 4.1.

Let a be a term ordering on k[X, Y] such that, t >a t' for t G k[X], t' G k[Y]. Let I be a zero-dimensional ideal in k[X, Y]. Consider an order ideal Oa{I} = Tn+m \ LTa{I}. Let O be an order ideal in k[X, Y] supporting a border basis of I, such that #(On k [Y ]) = #(Oa {I}n k [Y ]). If G is O-border basis of I, then G n k [Y ] is {On k [Y ])-border basis of I n k[Y ].

Proof. Observe that the border of On k [Y ] is dO n k [Y ]. Let dO n k [Y ] = {b,.....br} and On k [Y ] = {^.....tp}.

By the assumption on cardinality, Oa{I} n k[Y] = {s-|,..., sp} for some terms s^,..., sp G k[Y]. Let t be a term in k[Y]. If t G {s!.....sp}, then NFa,I(t) = t G (s-i.....sp)k.

Suppose t G {si,...,sp}. Then t = LTa(f) for some f G I. If a term t' G Supp(f) depends on xi for 1 < i < n, then t' >a t, which is a contradiction to t = LTa(f). Hence, f G k[Y]. Let f = t + h for h G k[Y] and LTa(h) <a t. Let f1.....fL

be the reduced Crobner basis of the Ideal I n k[Y] with respect to the term ordering o^^. Then f = h1f1 + • • • + hJi for h1,...,hi G k[Y]. We apply the Division Algorithm for Crobner bases to obtain NFoI(t). In the above situation we subtract a polynomial in k[Y] and the difference also belongs to k[Y]. At each step a reduction has the same properties, because every subsequent term is either in {si,..., Sp} or is a leading term of a polynomial in I n k[Y]. Finally, the rest has support in Oo{I} n k[Y]. Therefore NFoI(t) G ^.....sp)k.

Let O \ {t1,..., tp} = {tp+1,... ,tu}. We compute the O-border basis G of I according to the algorithm in Proposition 3.2. We calculate the matrix T of step (T3). NFoI(t[) = Yip=1 TjSj for 1 < i < p. Since O and Oo{I} are order ideals both supporting border bases of I, then #Oo{I} = u. Thus,

T11 • • T1p *

TP1 • • Tpp

0 • • 0 *

0 • •0

G MatuXu(k).

By Proposition 3.2 the matrix T is invertible. It follows from Gauss-Jordan algorithm that T 1 is of the form where A G Matpxp(k ) and C G Mat(u_p)x(u_p) (k ). Let dO \ {b1,..., br} = {br+1,...,bw}. Analogously to step (T3)

we obtain in step (T5) that B =

' A' *

, where A G Matpx(k) and C G Mat,

u~p)x(w~r]

)(k). Therefore T-1B =

(aij)1<i<u,1<j<w, where ay = 0 for p + 1 < i < u, 1 < j < r. By (T6), we have gj = bj — ajjtj for 1 < j < w. Hence 9j = bj — LP=1 aijti for 1 < j < r. Thus gj e k[Y] for 1 < j < r.

The set {g1,...,gr} is a border basis as a consequence of Proposition 3.1. Namely, S(gi,gj) e k[Y] for 1 < i,j < r. Moreover, if gi, gj are neighbors, then S(gi, gj) = c1g1 + • • • + cwgw for c1,..., cw e k. Suppose that there exists s > r such that cs = 0. Then at least one monomial cs0bs0 for s0 > r cannot be reduced, which is a contradiction. Hence

S(gi,gj ) = c1g1+ ••• + crgr. □

Theorem 4.1 provides a simple means for computing the border basis of the restriction of an ideal to a polynomial ring in a smaller number of variables. This is also the starting point for subsequent constructions.

An ideal I in k[X] is zero-dimensional if and only if I n k[xi] = (0) for every 1 < i < n (3.7.1 of [6]). Let A, B be zero-dimensional ideals. Then (A n B) n k[xj] = (A n k[xi]) n (B n [xj]). Since k[xj] is a principal ideal domain, then A n k[xj] = (f) and B n [xj] = (g) for nonzero polynomials f, g. Then fg e (A n k[xj]) n (B n [xj]). Hence, the intersection of zero-dimensional ideals is also zero-dimensional. The following elementary proposition is presented in [2].

Proposition 4.1.

Let A, B be ideals (not necessarily zero-dimensional) in k[X]. Let I < k[t,X] = k[t,x-\,... ,xn] be an ideal, such that, I = (tA, (t — 1)B). Then A n B = I n k[X}.

Therefore, to determine the border basis of the intersection of two zero-dimensional ideals, it suffices to compute the border basis of I and then apply Theorem 4.1. We only need I to be zero-dimensional. The latter is equivalent to ideals A and B having no common zero because V(I) = {0} x V(B) U {1} x V(A) U (k \ {0,1}) x (V(A) n V(B)). This example is a generic case, hence, algorithms of [4] work here in a generic case.

Note also that in [1] a generalization of the notion of border basis is introduced for positive dimensional ideals together with a corresponding algorithm. However the latter algorithm does not compute every border basis but only a border basis with respect to the specified ordering. For some applications, this is sufficient. Further development of the border basis theory for positive dimensional ideals may make the condition on the dimension not substantial.

Proposition 4.2 ([2], 3.2).

Let k[Y] = k[y■],..., yp], k[X] = k[x1,...,xn]. Let f : k[Y] — k[X] be a k-algebra homomorphism. If B = (y^ — f(y^).....yp — f(yp)) C k[X, Y], then ker f = B n k[Y]. □

Proposition 4.2 and Theorem 4.1 provide an algorithm for determining a border basis of the kernel of a homomorphism of polynomial rings (under usual assumption of dimension zero or applying results for positive dimensions where possible). It gives a border basis of the ideal of relations Aa) in the case where f1,...,fp are polynomials, because R C k[X]. As a consequence, we obtain information on rational constants of the associated factorizable derivation (see Section 2). We can give an algorithm also in the case where f1,...,fp are not polynomials, but it is quite laborious and it uses Grobner bases in one of the steps. In this case we don't know an algorithm based solely on border bases. We remark that the proof of Theorem 4.1 employs Grobner bases but the theorem itself does not. Hence, the algorithms derived from Propositions 4.1 and 4.2 are based purely on border bases.

References

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