Download STRATIFICATION BY THE LOCAL HILBERT

STRATIFICATION BY THE LOCAL HILBERT-SAMUEL
FUNCTION
ROUCHDI BAHLOUL
Abstract. The main goal of this article is to present a method for
computing a stratification by the local Hilbert-Samuel function for a
given polynomial ideal. The idea is to introduce a copy of the main
variables and, by a diagonal argument, reduce the work to a computation
of parametric standard bases relatively to a specific weighted order. The
difficulty being the same, we present a general method working with
arbitrary monomial orders, based on the use of a slight generalization
of pseudo standard bases. We present several variants, one of them
avoiding any irreducible or primary decomposition.
Introduction
Several authors contributed to simplify the monumental work by Hironaka: the resolution of singularities [Hir64]. Some of them worked on an
algorithm for it (see e.g. [Vil, BiMi, Hau, WÃlo]). The basic principle of
the proof by Hironaka consists in the existence of a sequence of blow-ups
associated with some local invariant. The first step consists in blowing up
the strata where the invariant is maximal. In Hironaka’s work this invariant is the Hilbert-Samuel function. Thus it needs the determination of the
strata of maximality of the local Hilbert-Samuel function. However, to our
knowledge, there does not exist an algorithm for this step that works in any
case.
The goal of this paper is the following: Given a polynomial ideal in a
computable field k of characteristic 0 (e.g. Q) which is a subfield of some
field K (e.g. C), to present an algorithm computing the whole stratification
over K by the local Hilbert-Samuel function.
Let us be more precise. Let I be an ideal in k[x] = k[x1 , . . . , xn ]. Given
x0 ∈ Kn , the local Hilbert-Samuel function of I at x0 over K is: r ∈
N 7→ HSI,x0 (r) = dimK (K[[x − x0 ]]/(K[[x − x0 ]]I + mrx0 )) where x − x0 =
(x1 − x0,1 , . . . , xn − x0,n ) and mx0 is the maximal ideal of the local ring
K[[x − x0 ]]. There exists a finite stratification of Kn into locally closed
subsets Kn = ∪W for which the following holds: For any W , HSI,x0 is
constant when x0 runs over W . The goal of this paper is to present an
algorithm that given generators of I, returns generators of ideals in k[x]
that define such W ’s and for each W returns the monomial ideal defining
the corresponding Hilbert-Samuel function.
Now let us describe our method. It is well know that given an ideal
I ⊂ k[x], we can compute the Hilbert-Samuel function at a given point by
using standard bases computations. The problem is that we don’t know
how to deal with a moving point. The idea is to fix the given point (e.g.
1
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ROUCHDI BAHLOUL
at 0), but as a counterpart we’ll need n supplementary indeterminates. Let
y = (y1 , . . . , yn ) be a new set of variables and let f1 , . . . , fq be the given
generators of I ⊂ k[x]. Define J as the ideal of k[x, y] generated by fi (x+y),
i = 1, . . . , q. Given x0 ∈ Kn , we denote by J|y=x0 the ideal of K[x] generated
by fi (x+x0 ). It is easy to check that HSI,x0 = HSJ|y=x0 ,0 . With this relation,
we are reduced to study the local Hilbert-Samuel function at the fixed point
x = 0 of a parametric ideal J ⊂ k[x, y] with parameters yi . Thus our method
shall use standard bases for parametric ideals.
The computation of the Hilbert-Samuel function uses a standard basis computation with respect to a specific local monomial order, that is
a weighted degree order where the weight is (−1, . . . , −1) (see [GrPf02b,
Def. 1.2.9]). However, in order to be as general as possible we shall adopt a
general approach by working with an arbitrary monomial order. When we
shall connect algebraic results to local results we will assume our order to
be local.
Let us give a simple example. Let I be the principal ideal generated by
f (x) = x21 + x32 . Let us write f (x + y) as a Taylor series:
f (x + y) = (y12 + y23 ) + (2y1 x1 + 3y22 x2 ) + (x21 + 3y2 x22 ) + (x32 ).
This expansion respects the total degree in x. Let us consider a (−1, −1)
degree order ¹ on the monomials in x, in other words an order that respects
the valuation in x. For ¹, the leading term of f (x + y) is 1 and the leading
coefficient is v = y12 + y23 . On C2 r {v = 0}, the Hilbert-Samuel function is
zero. Now let us work on the space {v = 0} (which is {f (y) = 0}). Here,
working modulo v, we can write
f (x + y) ≡ (2y1 x1 + 3y22 x2 ) + (x21 + 3y2 x22 ) + (x32 ).
Again we fix a monomial order on x as above. Notice that we have some
freedom: we can choose ¹ in order that the leading term is x1 or x2 with
the corresponding leading coefficients 2y1 or 3y22 . In fact it is the same since
we work along v = 0. Let us choose the leading monomial as 2y1 · x1 . Let us
denote V = {v = 0} and s = {(0, 0)} in C2 . We obtain that on V r s, the
Hilbert-Samuel function equals that of C[x1 , x2 ]/(x1 ). Finally, it remains s
(in fact we work modulo the ideal generated by v, 2y1 , 3y22 ) on which
f (x + y) ≡ x21 + x32 .
Finally, we get the stratification
C2 = (C2 r V ) ∪ (V r s) ∪ s
with constancy of the Hilbert-Samuel function on each strata. This is an
approximative exposition of the method in a special case (principal ideal
with an irreducible generator) but the basic idea is here.
To end the introduction le us describe the structure of the paper. In
section 1, we introduce standard bases in the algebraic setting following
[GrPf02b], then in the analytic and formal setting following [CaGr04] (in
fact we shall be more general than loc. cit.). In section 2, we establish
a result on specialization of parametric standard bases. To this end, we
propose three methods. The first one uses a slight generalization of pseudo
standard bases in [GrPf02b, GrPf02a]. The second one uses standard bases
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
3
and the third one uses Gröbner bases and a Lazard’s type homogenization.
In section 3, we propose a general algorithm for a stratification by a constant
set of exponents (or initial ideal) with three variants: one uses primary
decomposition and the two others do not. We apply this to our initial
problem on Hilbert-Samuel functions. We end this section with 3.4 where
additional finiteness results are presented.
Notations.
• k: a computable field of characteristic zero. K: a field containing k
(for some variants, K shall be algebraically closed).
• hf1 , . . . , fq i the ideal generated by the fi ’s.
• x = (x1 , . . . , xn ), y = (y1 , . . . , ym ): sets of variables.
• K{x}: (when K is a subfield of C) the convergent power series ring.
• Given an ideal I of k[y], VK (I) := {P ∈ Spec(K[y])|I ⊂ P} the
affine scheme defined by I over K.
• Given P ∈ Spec(K[y]), K(P) := Frac(K[y]/P).
Given c ∈ K[y], (c)P ∈ K(P) is the specialization of c to P. It is
the class of c in K[y]/P viewed in K(P).
Given f ∈ k[x, y] (or more generally in K[x, y]), (f )P ∈ K(P)[x] is
the specialization of f to P (here and below, we assume k ⊂ K).
Given an ideal J ⊂ k[x, y], (J)P is the ideal of K(P)[x] generated
by the (f )P ’s with f ∈ I.
Acknowledgements. I would like to thank Monique Lejeune-Jalabert and
Vincent Cossart for submitting to me the initial problem and many valuable
discussions, and Anne Frühbis-Krüger for a careful reading of the manuscript
and valuable suggestions.
1. Recalls and preliminaries on standard bases
Here we shall recall definitions and main results concerning standard
bases, firstly in the algebraic setting and secondly in the analytic and formal
setting. Our reference for the algebraic part is the book Singular [GrPf02b]
by Greuel and Pfister (see chapters 1 and 2). For the second part we shall
refer to part 1 of [CaGr04] by Castro-Jiménez and Granger entirely dedicated to standard bases for polynomials, analytic and formal power series.
We shall also refer to chapter 6 of [GrPf02b] for formal power series. For
the analytic and formal part, we shall adopt a more general approach than
in the references above. Indeed, we shall use arbitrary local orders instead
of special weighted degree orders. Let us precise that, actually working with
arbitrary local orders does not require any extra effort.
1.1. Monomial order and associated ring. As usual, if α ∈ Nn then xα
denotes xα1 1 · · · xαnn . A monomial order is a total order ¹ on the monomials
0
xα compatible with the product, that is: if xα ≺ xα then for any β, xα+β ≺
0
xα +β . We say ¹ is global if 1 is the minimal monomial; local if 1 is maximal;
otherwise ¹ is called mixed.
In the sequel, we will identify a monomial order with the induced order
on Nn .
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Given a matrix U ∈ GL(n, R), we define a monomial order ≤U : xα <U
xβ ⇐⇒ U α <lex U β where <lex is the lexicographical order on Rn . Robbiano [Ro85] has proven that any monomial order is equivalent to some
≤U .
Let A be a ring with inclusions of rings
P k[x]α ⊆ A ⊆ k[[x]] and let ¹ be a
monomial order. For f ∈ A, write f = α cα x as a power series expension.
We define the support of f as Supp(f ) = {α ∈ Nn |cα 6= 0}. When they
make sense, we define the leading exponent of f exp(f ) = max¹ Supp(f ),
the leading term lt¹ (f ) = xexp¹ (f ) , the leading coefficient lc¹ (f ) = cexp¹ (f )
and the leading monomial lm¹ (f ) = lc¹ (f )lt¹ (f ). These notions always
make sense if A = k[x]. If A = k[[x]], they always make sense if ¹ is local.
Now, let us fix a monomial order ¹. Let R = k[x]¹ be the localization of
k[x] with respect to the multiplicative set S¹ = {g ∈ k[x] r {0}| exp¹ (f ) =
0}. Notice that R = k[x] if (and only if) ¹ is global and R = k[x](0) , that
is the localization at 0, if (and only if) ¹ is local. In any case we have
inclusions of rings k[x] ⊆ R ⊆ k[[x]]. Thus the notations above apply to
the elements of R. In fact, if f ∈ R, and g ∈ S¹ satisfies gf ∈ k[x], then
exp¹ (f ) = exp¹ (gf ).
1.2. Standard bases in the algebraic situation. For simplification, we
shall forget the subscript ¹. For the moment A denotes either k[x] or
R = k[x]¹ . Let J be a non zero ideal of A. We define the set of leading
exponents Exp(J) = {exp(f )|f ∈ J r {0}}.
Definition 1.2.1. A Sfinite set G ⊂ A is called a standard basis of J if
G ⊂ J and Exp(J) = g∈G (exp(g) + Nn ).
A standard basis G is said to be minimal if 0 ∈
/ G and (exp(f ) ∈ exp(g) +
Nn ⇒ f = g) for any f, g ∈ G.
By Dickson lemma (see [GrPf02b, lemma 1.2.6]), a standard basis exists.
A minimal standard basis can be constructed by deleting some elements
from a given standard basis (see [GrPf02b, Rem. 1.6.3]).
Remark. Assume A = k[x]. If ¹ is global we say Gröbner basis instead
of standard basis. A Gröbner basis generates the ideal but a standard basis
does not in general.
From now on, A = R = k[x]¹ .
Definition 1.2.2 ([GrPf02b, Def. 1.6.4]). Let S(R) denote the set of finite
subsets of R. A map NF : R × S(R) → R, (f, G) 7→ NF(f |G) is called a
normal form if, for any f ∈ R and G ∈ S(R), we have
(0) NF(0|G) = 0
S
(1) NF(f |G) 6= 0 ⇒ exp(NF(f |G)) ∈
/ g∈G exp(g) + Nn
(2) There exist ag ∈ R and u ∈ R∗ = S¹ such
P that r := uf − NF(f |G)
has a standard representation: r =
g∈G ag · g, with exp(r) º
exp(ag g) for all g such that ag 6= 0.
(3) If {f } ∪ G ⊂ k[x] then the ag and u above can be taken in k[x].
Remark. In fact this is the definition of a polynomial weak normal form
in the terminology of [GrPf02b]. However we shall mainly use this notion
here.
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
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A normal form always exists: see [GrPf02b, 1.6, 1.7] with NFBuchberger
when ¹ is global and NFMora in general. NFMora is a variant of Mora’s
division [Mo82].
Lemma 1.2.3 ([GrPf02b, lemma 1.6.7]). Let J be an ideal of R, G be a
standard basis of J and NF be a normal form. For any f ∈ R, f ∈ J if and
only if NF(f |G) = 0.
Consequently, G generates J over R (but not over k[x] in general).
Definition 1.2.4. Let f, g be non zero elements in R. Set α = exp(f ),
β = exp(g) and γ = lcm(α, β), i.e. γi = max(αi , βi ). We define the Spolynomial (or S-function) of f and g as: S(f, g) := lc(g)xγ−α f −lc(f )xγ−β g
Theorem 1.2.5 ([GrPf02b, Th. 1.7.3]). Let J ⊂ R be an ideal and G a
finite subset of J. Let NF be a normal form (we may forget condition (3) in
Def. 1.2.2). The following are equivalent:
(1) G is a standard basis of J.
(2) NF(f |G) = 0 for any f ∈ J.
(3) Each f has a standard representation
P with respect to G that is: there
exist some ag ∈ R such that f = g∈G ag g with exp(f ) º exp(ag g)
for all g such that ag 6= 0.
(4) G generates J and for any g, g 0 ∈ G, NF(S(g, g 0 )|G) = 0.
The implication (4) ⇒ (1) is called Buchberger criterion.
1.3. Divisions with uniqueness - preparatory results. Here we shall
recall some known results and prove preparatory ones concerning divisions
and standard bases in k[x], C{x} and k[[x]]. We shall also make links with
the results of the previous section and be concerned with an extension field
k ⊂ K.
Originally, such divisions appeared in Hironaka [Hir64], Grauert [Gra72],
Buchberger [Bu70]. We shall follow more or less the presentation in [CaGr04,
chapt. 1].
Let R be one of the following: k[x], C{x}, k[[x]] and ¹ be a monomial
order. When R = k[x], ¹ shall be global, otherwise it shall be local. In the
latter case, the notions of leading exponent, monomial, etc make sense.
Let us now recall the division theorem with unique quotients and remainder as in [CaGr04]. For e1 , . . . , er in Nn , consider the following partition of
Nn .
• ∆1 = e1 +Nn and for j = 2, . . . , r, ∆j = (ej +Nn )r(∆1 ∪· · ·∪∆j−1 ).
¯ = Nn r (∆1 ∪ · · · ∪ ∆r ).
• ∆
Theorem 1.3.1 ([CaGr04, Th. 1.5.1]). Let g1 , . . . , gr be non zero elements
in R and consider the partition associated with the exp(gj ). For any f ∈ R,
there exists a unique (q1 , . . . , qr , R) ∈ Rr+1 such that:
(i) f = q1 g1 + · · · + qr gr + R
(ii) for any j, if qj 6= 0 then Supp(qj ) + exp(gj ) ⊂ ∆j
¯
(iii) if R 6= 0 then Supp(R) ⊂ ∆.
R is called the remainder of the division of f by g1 , . . . , gr (w.r.t. ¹).
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Because of the partition of Nn , we have:
¯
exp(f ) = max≺ {exp(q1 g1 ), . . . , exp(qr gr ), exp(R)} and exp(R) ∈ ∆.
Remark. In the case R = k[x], the map (f, G = {g1 , . . . , gs }) 7→ R is a
normal form in the sense of Def. 1.2.2. The division in Th. 1.3.1 is more
rigid than the notion of normal form (because without any choice in the
division process) but it is sometimes more easy to work with.
The statement of Theorem 1.3.1 is not verbatim the same as [CaGr04,
Th. 1.5.1]. We will show how we can recover our result from loc. cit. but
first, let us make a remark:
Remark 1.3.2. Let ¹0 be any monomial order such that exp¹ (gj ) = exp¹0 (gj ),
then any division relative to ¹ is a division relative to ¹0 (i.e. shall output
the same qj ’s and the same R).
Lemma 1.3.3. Let ¹ be a local monomial order. Let U = (u1 , . . . , un )t ∈
GL(n, R) be a matrix representing ¹ following Robbiano (here, ()t is the
transpose, thus ui are the rows of U ). Given a finite set G in k[[x]], there
exists a small pertubation ε ∈ (R≤0 )n such that u1 + ε ∈ (Q<0 )n and if we
denote by U 0 = (u1 + ε, u2 , . . . , un )t , we have exp¹U (g) = exp¹U 0 (g) for any
g ∈ G.
Proof. We just apply Prop. 8 in [ACG01] to elements of k[[x]]. Indeed,
k[[x]] is a subring of homogeneous elements of D̂[t] with the notations of
[ACG01] and any monomial order on xα can be extended to an order as in
loc. cit.
¤
Now let us explain how Th. 1.3.1 is consequence of [CaGr04, Th. 1.5.1].
Proof. Let us do it in the local case (for the global case, it is the same). In
loc. cit. the authors proved the division theorem for a specific order ≤u
defined in this way: they fix a weight vector (or a linear form) u ∈ Qn with
ui ≤ 0 (in fact they work with the ui ’s ≥ 0 but they consider the minimum
instead of the maximum) and a global order ≤ and they define ≤u in a
lexicographical way by u and the inverse of <.
From lemma 1.3.3 and Remark 1.3.2, it is easy to see that we are reduced
to the hypotheses of [CaGr04, Th. 1.5.1].
¤
In the case of power series, we can define the notion of standard basis in
a similar way to the algebraic case (and it exists by Dickson lemma again).
Under the assumptions of the division theorem, the following holds.
Proposition 1.3.4 ([CaGr04], Cor. 1.5.4). Let G = {g1 , . . . , gr } in J, these
two statements are equivalent:
• For any f ∈ J, the remainder of the division of f by G is zero.
• The set G is a standard basis of J.
Defining S-functions as in Def. 1.2.4, we have a Buchberger criterion:
Again, under the assumptions of Th. 1.3.1, we have:
Proposition 1.3.5 ([CaGr04], Prop. 1.6.2). Let G = {g1 , . . . , gr } be a set
of generators of an ideal J ⊂ R. Then G is a standard basis of J if and
only if: for any j, j 0 , the remainder of the division of S(gj , gj 0 ) by G is zero.
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
7
Let us make a remark that shall be used in the sequel. Let k ⊂ K be
two fields. Set R = R(k) and denote by R(K) the ring R(k) where k is
replaced with K.
Remark 1.3.6. Suppose we are given g1 , . . . , gr , f ∈ R(k) and ¹ an order
as in Th. 1.3.1. The ¹-division of f by G in R(k) and in R(K) are the
same, i.e. gives the same outputs qj and R. Therefore, by Prop. 1.3.5, a
standard basis of an ideal in R(k) is also a standard basis of the extented
ideal in R(K). The same arguments imply that a standard basis of J ⊂ C{x}
is a standard basis of C[[x]]J.
Global-local link and extension results
Our main result is local. However the constructions shall be global (or at
least algebraic), so we need a result to connect global and local setting.
Proposition 1.3.7. Let k ⊆ K be two fields. Let ¹ be a local order on Nn
and J ⊂ k[x] be an ideal. If G ⊂ J is a ¹-standard basis of J then it is a
¹-standard basis of K[[x]] · J.
Proof. First let us assume that G generates J. Take two elements g, g 0
in G and consider the S-function S(g, g 0 ), it belongs to J. Now consider
the division of S(g, g 0 ) by G in K[[x]]. At each step of the division, every
thing is in k[x] and thus the leading exponent of the partial remainder is
in Exp¹ (J), therefore the remainder is necessarily zero. We then conclude
with Buchberger criterion 1.3.5.
Now assume that G does not generate J. Let G0 be any system of generators of J. The arguments above applied to G ∪ G0 imply that G ∪ G0
is ¹-standard basis of K[[x]]J. Since G is a standard basis of J, for each
g 0 ∈ G0 , exp¹ (g 0 ) ∈ ∪g∈G (exp¹ (g) + Nn ) thus G is ¹-standard basis of
K[[x]]J.
¤
In one of the variants of the algorithm, we shall use the following.
Proposition 1.3.8. Assume that k ⊂ K and K is algebraically closed. Let
Q ⊂ k[y]
√ be an ideal and h ∈ k[y] then:
h∈
/ Q ⇐⇒ VK (Q) r VK (h) 6= ∅.
Here y = (y1 , . . . , ym ) as in the introduction.
Proof. Assume that the set above is empty, then by Hilbert’s Nullstellensatz,
hN ∈ K[y]Q for some N ∈ N. Let ¹ be a global order and G be ¹-Gröbner
basis of Q. By Rem. 1.3.6, it is also a ¹-Gröbner
basis of K[y]Q. Consider
P
the division of hN by G in k[y]: hN = j qj gj + R. This division is also
the division of hN by G in K[y], and since hN ∈ K[y]Q, the remainder R
is zero which implies that hN ∈ Q. We have proved the implication ”⇒”.
Since ”⇐” is trivial, the proof is complete.
¤
√
Notice that there exists a relatively easy way to check if h ∈ Q (see
e.g. §1.8.6 in [GrPf02b]).
Here ”relatively easy” means without computing
√
a Gröbner basis of Q.
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ROUCHDI BAHLOUL
1.4. Hilbert-Samuel function and Standard basis. Denote by m the
maximal ideal in C{x} (or more generally K[[x]]). For an ideal I in C{x},
the (local) Hilbert-Samuel function of I HSI : N → N is defined by HSI (r) =
dimC (C{x}/(I + mr+1 )). For a set E ∈ Nn such that E + Nn = E, we define
its Hilbert-Samuel function HSE : N → N as HSE (r) = card{α ∈ Nn ; α ∈
Nn r E, |α| ≤ r}.
0
Lemma 1.4.1. Let ¹ be a local order such that: |α| < |α0 | ⇒ xα Â xα .
Then HSI = HSExp¹ (I) .
Proof. See e.g. [GrPf02b, Prop. 5.5.7] or [CaGr04, Prop. 1.9.3].
¤
2. Results on parametric standard bases
We recall that k is a computable field of characteristic zero and K is a
field containing k.
Let us consider an ideal J in k[x, y] where the system of variables y =
(y1 , . . . , ym ) is seen as a parameter. In this section, we fix ¹ to be a monomial
order on the terms xα but ¹ is taken arbitrary, i.e. we don’t suppose it to
be local. The goal of this section is, given an ideal Q in k[y] (not necessarily
prime), to present a (finite) algorithm for computing a finite set G ⊂ J +
k[x, y]Q and h ∈ k[y] r Q such that for any P ∈ VK (Q) r VK (h), (G)P is a
¹-standard basis of (J)P and ∪g∈G (exp≺ ((g)P ) + Nn ) is independent of P.
(Notice that with little more efforts we could construct G ⊂ J.)
We shall present three variants for the algorithm.
The first one, treated in §1 shall use pseudo standard bases following
Greuel and Pfister ([GrPf02b, Ex. 2.3.6-2.3.8] and [GrPf02a]). Our way is
slightly different in the sense that only the case Q = (0) is treated in loc.
cit. From a more theoretical point of view, we may apply the results of loc.
cit. to the ring k[y]/Q (for instance by reducing each output modulo Q)
and this is more or less what will happen.
The second variant, given in §2, uses usual standard bases.
The third variant, treated in §3, uses global orders and a homogenization
as in Lazard [La83].
We think it is interesting to have a choice between several methods. Moreover, not all the computer algebra programs can work with pseudo standard
bases. Some of them work only with global orders.
2.1. The main result for parametric standard bases. Here are the
data: an ideal J ⊂ k[x, y], a field k of characteristic zero, a field K ⊇ k, an
ideal Q ⊂ k[y], a monomial order ¹ on the xα ’s.
Assumption: Assume that J * k[x, y] · Q.
Proposition
2.1.1. There exists a finite algorithm for computing G ⊂ J
Q
and h = f inite hi with hi ∈ k[y] r Q such that for any P ∈ VK (Q) r VK (h)
we have:
• (G)P is a standard basis of (J)P (and then of Frac(K[y]/P)[x]¹ (J)P ).
• For each g ∈ G, exp((g)P ) is independant of P (thus so is Exp((J)P )).
√
We noticed (see 1.3.8) that if K is algebraically closed and h ∈
/ Q then
VK (Q) r VK (h) is not empty. In particular, this is true if Q ⊂ k[y] is prime.
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
9
2.2. First method: an analogue of pseudo standard bases. Let C be
an integral commutative ring with 1. For our problems C shall be k[y]. We
fix a monomial order ¹ on the monomials xα . In C[x], we can define the
notion of leading exponent, leading coefficient (in C), etc. Set S¹ = {f ∈
−1
C[x]| exp(f ) = 0 and lc(f ) = 1} then define R = S¹
C[x] as the localization
w.r.t. S¹ .
Definition 2.2.1 (See [GrPf02a] or [GrPf02b, pages 124-125]).
• A pseudo normal form NF is defined in the same way in Def. 1.2.2
except that u in (2), instead of being a unit, is an element of R such
that lm(u) is product of some lc(g), g ∈ G.
• Given an ideal J ⊂ R and a pseudo normal form NF, a pseudo
standard basis is a finite set G ⊂ J satisfying:
(1) G generates J over R.
(2) For any g, g 0 ∈ G, NF(S(g, g 0 )|G) = 0.
Pseudo normal forms exist (NFMora in [GrPf02b] is one) and pseudo
standard bases also (see Alg. 2.3.8 in [GrPf02b]).
Remark 2.2.2.
(1) It is important to notice that both the notion of
pseudo normal form and the algorithm NFMora make sense even if
the ring C is not integral. However Prop. 3.2 in [GrPf02a] (see also
Exerc. 2.3.7 in [GrPf02b]) is not true as C is not integral.
(2) We shall see that the property of being a pseudo standard basis does
not depend on the choice of a pseudo normal form.
Now let us generalize these constructions. In the sequel Q ⊂ C is a given
ideal, not necessarily prime. Given f ∈ R = C[x]¹ , we define expmodQ (f ) :=
exp(f modQ), where f modQ means the class of f in C/Q[x]¹ . We define naturally ltmodQ (f ). Then lcmodQ (f ) denotes the coefficient (in C) of ltmodQ (f )
in the expension of f , finally lmmodQ (f ) := lcmodQ (f )ltmodQ (f ).
Now for an ideal J ⊂ R, we define ExpmodQ (J) = {expmodQ (f )|f ∈
J r RQ}.
Definition 2.2.3. A pseudo standard
S basis of J relatively to Q is a finite
set G ⊂ J such that ExpmodQ (J) = g∈G (expmodQ (g) + Nn ).
Remark. Such a set exists by Dickson lemma but our goal is to show how
it can be constructed under the hypothesis of Prop. 2.1.1. Notice that if
Q = (0), this notion coincides with the usual notion of pseudo standard
basis.
Definition 2.2.4. A pseudo normal form NF relatively to Q is a map NF :
R × S(R) → R, (f, G) 7→ NF(f |Q G) such that for any f ∈ R and G ∈ S(R),
we have
(0) NF(q|Q G) ∈ Q for all q ∈ RQ
S
(1) NF(f |Q G) ∈
/ RQ ⇒ expmodQ (NF(f |Q G)) ∈
/ g∈G (expmodQ (g) + Nn )
(2) There exist ag ∈ R, q ∈ RQ, and u ∈ R such that lmmodQ (u) is a
product of some lcmodQ (g),Pg ∈ G for which we have
r := uf − NF(f |Q G) = g∈G ag g + q
with expmodQ (r) º expmodQ (ag g) for all g such that ag 6= 0.
10
ROUCHDI BAHLOUL
(3) If {f } ∪ G ⊂ C[x] then the ag and u and q above can be taken in C[x].
We define the S-polynomial modulo Q: S modQ (f, g) by using expmodQ ,
lcmodQ etc instead of exp, lc, etc.
As for standard bases, we have a characterization of pseudo standard
bases in terms of pseudo normal forms and S-polynomials.
Proposition 2.2.5. Let J ⊂ R be an ideal and G a finite subset of J. Let
NF be a pseudo normal form. The following are equivalent:
(1) G is a pseudo standard basis of J relatively to Q.
(2) NF(f |Q G) ∈ Q for any f ∈ J.
(3) For any f ∈ J, there exists ag ∈ R for all g ∈ G, q ∈ RQ, and
u ∈ R withP
lmmodQ (u) being a product of lcmodQ (g) (g ∈ G) such
that: uf = g∈G ag g + q with expmodQ (f ) º expmodQ (ag g) for all g
such that ag 6= 0.
(4) For any f ∈ J, there exists u as above such that uf ∈ RG + RQ and
for any g, g 0 ∈ G, NF(S modQ (g, g 0 )|Q G) ∈ Q.
The proof is analogous to that of Th. 1.7.3 in [GrPf02b]. In fact, the
implications (1) ⇐⇒ (2) ⇐⇒ (3) ⇒ (4) are relatively easy to prove and
the last implication shall not be used here but we think it is interesting to
state it.
A slight generalization of the notion of écart is necessary to go further.
Given f ∈ C[x], we define the écart modQ: écartmodQ (f ) := écart(f modQ) =
deg(f modQ) − deg lt(f modQ).
Algorithm 2.2.6 (NFMora(•|• •)).
Input: f ∈ C[x], G a finite set in C[x], Q ⊂ C an ideal.
Output: h = NFMora(f |Q G) ∈ C[x] a pseudo normal form of f w.r.t. G
relatively to Q.
• h := f ;
• T := G;
• While (h ∈
/ C[x]Q and Th := {g ∈ T such that ltmodQ (g)|ltmodQ (h)} =
6 ∅)
choose g ∈ Th with écartmodQ (g) minimal;
If (écartmodQ (g) > écartmodQ (h)) then T := T ∪ {h};
h := S modQ (h, g);
• Return h.
All the definitions were made in order to have the following equality:
NFMora(f |Q G) = NFMora(f modQ|GmodQ). This proves both termination and correctness of this algorithm.
Remark 2.2.7. From an algorithmic point of view all the ”notions” modQ
(as expmodQ , écartmodQ , etc) can be ”done” in a finite number of steps in
our situation: Q ⊂ C = k[y]. For example given f ∈ k[x, y], how can we
compute expmodQ (f )? We consider a monomial order on the y β , say ≤0 and
compute a standard or Gröbner basis of Q, say G0 . Then we consider a
monomial order, say ≤, on the monomials xα y β whose restriction to y β is
≤0 (for example the block order (¹, ≤0 )). Then obviously, G0 is a standard
basis of k[x, y]Q w.r.t. ≤. Given f ∈ k[x, y], we compute a normal form
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
11
r = NF≤ (f |G0 ) and we get expmodQ
(f ) = exp¹ (r).
¹
Now to end this paragraph, let us return to the situation: J ⊂ k[x, y],
C = k[y]. Let G be a system of generators of J.
Claim 2.2.8.
(1) The output G of the algorithm Standard (see [GrPf02b,
Algo. 1.7.1]) applied to G and a pseudo normal form relatively to
Q, e.g. NFMora(− |Q −), is a pseudo standard basis of J relatively
to Q
Q
(2) Let G be pseudo standard basis and let h = g∈G lcmodQ (g) then G
and h satisfy Prop. 2.1.1.
Proof. The proof of (1) follows from the algorithm Standard and Prop. 2.2.5.
Let us show (2). The proof is based on the fact that NFMora gives rise to
a normal form when it is used with a field k and to a pseudo normal form
when it is used with a ring C. Notice also that NFMora acts the same if we
work with the field extension K ⊇ k.
For any P ∈ Spec(K[y]) such that h ∈
/ P ⊃ Q, the following holds:
(lmmodQ (g))P = lm((g)P ) and thus (S modQ (g, g 0 ))P = S((g)P , (g 0 )P ), for
any g 6= g 0 ∈ G. As a consequence, if we follow step by step NFMora, we obtain that (NFMora(S modQ (g, g 0 )|Q G))P is equal to NFMora(S((g)P , (g 0 )P )|(G)P ).
Since the former is zero by Prop. 2.2.5(4), the latter is so.
Since (G)P generates (J)P by Prop. 2.2.5(4), it follows from Buchberger’s
criterion (Th. 1.2.5) that (G)P is a standard basis of (J)P .
¤
2.3. Second method: using standard bases. Denote by J˜ = J+k[x, y]Q.
Let ≤0 be a monomial order on the y β . In practice it is better that ≤0 be a
global order but things work even if it is not the case. Define a block order
on xα y β as ≤= (¹, ≤0 ), here ¹ is the monomial order on xα fixed at the
begining of this subsection.
Note. For an element f ∈ k[x, y], we will work with two types of leading exponents (and of leading terms, coefficients, etc): exp¹ (f ) ∈ Nn and
exp≤ (f ) ∈ Nn+m .
Remark 2.3.1. For any f ∈ k[x, y], exp≤ (f ) = (exp¹ (f ), exp≤0 (lc¹ (f ))).
Let G be a standard basis of J˜ = J + k[x, y] · Q w.r.t. ≤. We shall
assume that G is minimal (a posteriori we shall see that minimality is not
necessary).
Lemma 2.3.2. For any g ∈ G: lc¹ (g) ∈ Q ⇐⇒ g ∈ Q.
Proof. The right-left implication is trivial. Let us prove the converse one.
Write g = lc¹ (g)xα +· · · where α = exp¹ (g). We have exp≤ (g) = exp≤ (lc¹ (g))+
˜ we have exp≤ (lc¹ (g)) ∈ exp≤ (g 0 ) + (α0 , β 0 )
(α, 0) but since lc¹ (g) ∈ Q ⊂ J,
for some g 0 ∈ G and (α0 , β 0 ) ∈ Nn+m . By minimality of G, we must have
(α, 0) = (α0 , β 0 ) = (0, 0). Therefore g = lc¹ (g).
¤
Lemma 2.3.3. G ∩ k[y] is a ≤0 -Gröbner basis of J˜ ∩ k[y].
This is beacause we have a block order. Details are left to the reader.
12
ROUCHDI BAHLOUL
Proposition 2.3.4. The set G = G r Q is a ¹-pseudo standard basis of J˜
relatively to Q.
Proof. Let us treat two cases.
Case 1: Q is included in but different from J˜∩ k[y]. By the previous lemma,
there exists necessarily a non zero g ∈ G ∩ k[y], that is expmodQ
(g) = 0.
¹
Thus G satisfies Def. 2.2.3.
Case 2: Q = J˜∩ k[y]. By the lemma above, G ∩ k[y] = G ∩ Q is ≤0 -standard
basis of Q.
˜ we may assume
Take f ∈ J˜ such that f ∈
/ k[x, y]Q. Since Q ⊂ J,
modQ
(f ) = lc¹ (f ).
lc¹
Write f = lc¹ (f )lt¹ (f ) + v. Let c = NF(lc¹ (f )|G ∩ Q) w.r.t. ≤0 . Since
G∩Q is a standard basis, r is not zero. Set f˜ = c·lt¹ (f )+v. By construction,
f˜ ∈ J˜ and exp¹ (f ) = exp¹ (f˜). By definition of G, there exists g ∈ G such
that exp≤ (f˜) ∈ exp≤ (g) + Nn+m .
By Remark 2.3.1, this implies exp≤0 (c) ∈ exp≤0 (lc¹ (g)) + Nm . Assume
by contradiction that g is in Q then by Lemma 2.3.2, g = lc¹ (g). Thus
exp≤0 (c) ∈ exp≤0 (g) + Nm but this is impossible since c is a normal form
w.r.t G ∩ Q. Thus such a g cannot be in Q, it is then in G. Finally, we
proved that G satisfies Def. 2.2.3.
¤
By using Claim 2.2.8(2) we conclude this second method.
Remark. We worked with J˜ instead of J so G is pseudo standard basis of
˜ not of J. Firstly, if one keeps in memory the computations done, it is
J,
not difficult to obtain a pseudo standard basis of J. Secondly, we notice
that having a pseudo standard basis of J˜ is in fact enough for our concern.
˜
Indeed, what is important is to have ExpmodQ (J) which equals ExpmodQ (J)
Q
modQ
and to have g∈G lc¹
(g).
2.4. Third method: Lazard’s type homogenization and Gröbner
bases. Let us fix any global order ≤0 on the y β , β ∈ Nm . Let z be a new
variable and let us define(the orders ¹z and ≤z :
|α| + k < |α0 | + k 0
0
0
xα z k ≺z xα z k ⇐⇒
0
or equality and xα ≺ xα ,
(
0
0
xα z k ≺z xα z k
0 α0 k 0
β
α
k
z
β
y x z < y x z ⇐⇒
0
or equality and y β <0 y β .
Here above, α, β, k are in Nn , Nm and N respectively. Note that these
orderings are global orders.
P
For any ring A and any f in A[x], write f = α cα xα and define the
P
homogenization of f in A[x, z] as h(f ) = α cα xα z d−|α| where d is the total
degree
P of f in the variables xi . More generally an element f of the form
f = α,k cα,k xα z k with cα,k = 0 if |α| + k 6= d is said to be homogeneous
(of degree d).
Now let fj , j = 1, . . . , q be the given generators of J ⊂ k[x, y]. Consider
0
a total degree order ¹T on the xα (i.e. such that |α| < |α0 | ⇒ xα ≺T xα )
then define ≤T as a block order (¹T , ≤0 ) on the xα y β .
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
13
Notice that a ≤0 -Gröbner basis of Q is a ≤T -Gröbner basis of k[x, y]Q
(it is a direct consequence of Buchberger’s criterion).
For each fj , let fj0 be a ≤T -normal form of fj w.r.t. any Gröbner basis of
Q
k[x, y]Q (in fact we can take a ≤0 -Gröbner basis of Q). Put h0 := j lc¹ (fj0 ).
Claim. For each j, lc¹ (fj0 ) ∈ k[y] r Q.
P
Now let G be a minimal ≤z -Gröbner basis of J 0 := q1 k[x, y, z]h(fj0 ) +
k[x, y, z]Q made Q
of homogeneous elements. Set G = {f|z=1 |f ∈ G r Q}.
Finally put h := g∈G lc¹ (g).
Proposition 2.4.1. For any P ∈ VK (Q) r VK (hh0 ), the conclusion of Prop.
2.1.1 holds.
Remark. Since J 0 is generated by homogeneous elements and Buchberger
algorithm conserves homogeneity, it is always possible to construct G made
of homogeneous elements.
Lemma 2.4.2.
(a) The claim above is true.
P
(b) For any P ∈ VK (Q), (J)P = qj=1 K(P)[x] · (fj0 )P .
P
(c) For any P ∈ VK (Q) r VK (h0 ), (J 0 )P = qj=1 K(P)[x, z] · h((fj0 )P ).
Proof. (a): Assume by contradiction that lc¹ (fj0 ) ∈ Q for some j. By
definition of ≤T , we have exp≤T (fj0 ) = exp≤T (lc¹T (fj0 ) · lm¹T (fj0 )). But
lc¹T (fj0 ) · lm¹T (fj0 ) is in k[x, y]Q. This contradicts the fact that fj0 is a ≤T normal form. Statement (b) is trivial. Let us prove (c). By definition of ≤T
and of h0 , it is clear the total x-degree of fj0 coincides with that of (fj0 )P . As
a consequence: (h(fj ))P = h((fj )P ) and statement (c) follows.
¤
LemmaP
2.4.3. For any P ∈ VK (Q) r VK (hh0 ), (G r Q)P is a ¹z -standard
basis of qj=1 K(P)[x, z] · h((fj )P ) and is made of homogeneous elements.
Proof. By the previous lemma, since P ∈ V (Q) r V (h0 ), (G r Q)P generates
the ideal in question. We conclude with Proposition 2.3.4.
¤
The following lemma is a classical result (see e.g. Exerc. 1.7.6 in [GrPf02b]).
Lemma 2.4.4. Let I be an ideal in k[x] generated
by f1 , . . . , fq . Let G be a
P
z
homogeneous ¹ -standard basis of the ideal j k[x, z]h(fj ). Then G|z=1 is
a ¹-standard basis of I. (In fact, any element of I has a ¹-standard over
k[x].)
Now the proof of Prop. 2.4.1 is a direct application of this lemma to
lemma 2.4.3.
3. Stratification with respect to a constant Exp, Application
to local Hilbert-Samuel function
Given J ⊂ k[x, y] and K ⊃ k as in the introduction (K is not necessarily
algebraically closed) and ¹ a monomial order on the terms xα , the goal of
this part is to describe (and justify) an algorithmic way for the following.
Claim. There exists a stratification Km = (∪Wk ) ∪ Z, where each Wk is a
locally closed subset of Km defined as the difference of the zero sets of two
ideals in k[y] and Z is a Zariski closed set defined by an ideal of k[y], with the
14
ROUCHDI BAHLOUL
following properties: For each k there exists a finite set Gk ⊂ k[x, y] such that
for any P ∈ Wk of SpecK (K[y]), Gk is ¹-standard basis of (J)P ⊂ K(P)[x];
and for any P ∈ Z, (J)P = (0).
Remark. As we noticed in the previous section, it is possible with some
extra care to construct each Gk in J. The union G of the Gk would be a
“comprehensive standard bases” in the sense that for any P ∈ Spec(k[y]),
(G)P is standard basis of (J)P ⊂ K(P)[x].
We are going to describe a construction by induction on the step l. At
each step l, we shall construct the following objects:
• A finite list Wl of couples (W, G) where W is a locally closed set
defined as the difference of the zeroset of two ideals of k[y], and G is
a finite set in k[x, y]. This list may be empty,
• A list (which may be empty) Ql of ideals Q of k[y],
• An ideal Il of k[y].
with the following properties:
S
S
(p1) Km = ( (W,G)∈Wl W ) ∪ VK (Il ) ∪ ( Q∈Ql Q),
(p2) For any (W, G) ∈ Wl , and for any P ∈ W of SpecK (K[y]), (G)P is
¹-standard basis of (J)P ⊂ K(P)[x].
(p3) J ⊂ k[x, y]Il . Thus, J reduces to zero along VK (Il ).
At step 0, we set W0 = ∅ and Q0 = {(0)} and I0 = h1i.
Now let us describe how to make the construction at step l + 1. Put
Wl+1 = Wl , Il+1 = Il , Ql+1 = Ql .
3.1. First variant: with primary decomposition. In this first variant,
we assume K to be algebraically closed. Each Ql shall be made of prime
ideals of k[y]. For each Q ∈ Ql , do the following:
(1) If J is not included in k[x, y]Q then:
Apply Proposition 2.1.1 to Q.
As outputs we obtain G ⊂ k[x, y] and h ∈ k[y] r Q.
By e.g. a primary decomposition and the computation of radicals,
find Q1 , . . . , Qs prime ideals of k[y] such that
(?)
VK (Q + hhi) = VK (Q1 ) ∪ · · · ∪ VK (Qs ).
We have VK (Q) = (VK (Q) r VK (h)) ∪ VK (Q + hhi) and
Q $ Qi .
Put Wl+1 := Wl+1 ∪ {(VK (Q) r VK (h), G)};
Ql+1 := (Ql+1 r {Q}) ∪ {Q1 , . . . , Qs }.
(2) If J is included in k[x, y]Q, then:
put Ql+1 := Ql+1 r {Q},
Il+1 := Il+1 ∩ Q.
Remark 3.1.1.
• Checking whether J ⊆ k[x, y]Q needs only to reduce
a system of generators of J with respect to a Gröbner basis of Q
(computed in k[y]). It is a part of the first method for Prop. 2.1.1
(see Remark 2.2.7). In any of the three methods of section 2, we
need to compute a Gröbner basis of Q.
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
15
• Since Q is prime and h ∈
/ Q, VK (Q) r VK (h) is a dense open set of
VK (Q).
• Relation (?) holds because if I1 , I2 are two ideals of k[y] then (K[y]I1 )∩
(K[y]I2 ) = K[y](I1 ∩ I2 ). Indeed, the computation of K[y]I1 ∩ K[y]I2
can be done by a Gröbner basis computation. It is the same computation that outputs I1 ∩ I2 (see e.g. [GrPf02b, §1.8.7]).
• Notice that the arguments can be shortened if Q is principal because
in this case, an irreducible decomposition over k[y] of the generator
of Q can be used instead of a primary decomposition.
It is clear that properties (p1), (p2), (p3) are satisfied for all l ∈ N. Let
us explain that for some l, Ql = ∅, this will prove both termination and
correctness. By construction, the Krull dimension of ideals in Ql decrease.
Therefore, for some l, all the Q’s in Ql become h1i or at least satisfy condition
(2). Thus we have at most m + 1 steps (recall that m is the number of yi ).
3.2. Second variant: without primary decomposition. Here we describe a general algorithm which does not use primary decomposition or
irreducible decomposition of varieties. Moreover, K is not assumed to be
algebraically closed.
For each Q ∈ Ql , do the following:
(1) If J is not included in k[x, y]Q then:
Apply Proposition 2.1.1 to Q.
As outputs we obtain G ⊂ k[x,
Q y] and a finite number of hi ∈ k[y]rQ,
say i = 1, . . . , r; and h := hi .
(1.a) If h ∈ Q:
We have VK (Q) = ∪i VK (Q + hhi i), and
(1)
Q $ Q + hhi i.
Put Ql+1 := (Ql+1√r {Q}) ∪ {Q + hhi i|i = 1, . . . , r}.
(1.b) If h ∈
/ Q and h ∈ Q:
We have VK (Q) = VK (Q + hhi) and
(2)
Q $ Q + hhi.
Put Ql+1 := (Ql+1√r {Q}) ∪ {Q + hhi}.
(1.c) If h ∈
/ Q and h ∈
/ Q:
We can write VK (Q) = (VK (Q) r VK (h)) ∪ VK (Q + hhi) and we
have
(3)
Q $ Q + hhi.
Put Wl+1 := Wl+1 ∪ {(VK (Q) r VK (h), G)};
Ql+1 := (Ql+1 r {Q}) ∪ {Q + hhi}.
(2) If J is included in k[x, y]Q, then:
put Ql+1 := Ql+1 r {Q},
Il+1 := Il+1 ∩ Q.
This achieves the construction for step l + 1.
Remark 3.2.1.√Notice that checking whether h ∈
computation of Q, see e.g. [GrPf02b, §1.8.6].
√
Q does not require the
16
ROUCHDI BAHLOUL
Again, it is enough to see that for some l, Ql is empty. The key point
is that in any case 1.a, 1.b, 1.c, we get a strict inclusion of ideals, that is
Q is included in some other ideal(s). The latters then become elements of
Ql+1 , and so on. So we get a tree made of strict inclusion of ideals. What
condition make a branch stop growing? If condition (2) (i.e. J ⊂ k[x, y]Q)
is satisfied then it stops, otherwise it is because of noetherianity of k[y].
If we are in condition (1), then by construction the r.h.s (right hand side)
ideal in equations (1, 2, 3) is always strictly bigger than the l.h.s. one. Thus
noetherianity implies that for some l, the r.h.s. is h1i = k[y]. Because of
(2), Ql becomes empty for some large enough l.
3.3. A third variant. Here we expose another variant
√ without primary
decompostion and also avoiding to check whether h ∈ Q. In this variant,
we replace (1.b, 1.c) by (1.a’) as follows:
(1) · · ·
(1.a) If h ∈ Q: · · ·
(1.a’) If h ∈
/ Q:
We can write VK (Q) = (VK (Q) r VK (h)) ∪ VK (Q + hhi) and we
have
(4)
Q $ Q + hhi.
Put Wl+1 := Wl+1 ∪ {(VK (Q) r VK (h), G)};
Ql+1 := (Ql+1 r {Q}) ∪ {Q + hhi}.
(2) · · ·
√
The difference with the second variant occurs if h ∈
/ Q and h ∈ Q. In
this case, VK (Q)rVK (h) is empty, so we add to Wl+1 the empty set coupled
with some useless G. Correctness and termination hold for the same reasons.
3.4. The algorithm concerning Hilbert-Samuel function. First, let
us make a remark. Applying the algorithm of section 3, we first obtain a
usually finer stratification than the one by a constant Exp. Comparing the
values of Exp on different strata and forming unions of appropriate strata,
we obtain the desired stratification.
Now let us apply the algorithm to get a local Hilbert-Samuel stratification.
3.4.1. Let J = J(x, y) ⊂ k[x, y] be an ideal and K a field containing k. Let
¹ be a (local) order on the terms xα as in Lemma 1.4.1. Apply one of the
three algorithms above, we can construct a stratification Km = (∪k Wk ) ∪ Z
such that on Z J reduces to zero and on each Wk the local Hilbert-Samuel
function of (J)P at x = 0 is constant.
3.4.2. Now let us go back to the original question. Take I ⊂ k[x] generated
by f1 , . . . , fq . Let us consider a new set of vaiables y = (y1 , . . . , yn ) and define
the ideal J = J(x, y) ⊂ k[x, y] with f1 (x + y), . . . , fq (x + y).
It is easy to check that given x0 ∈ Kn , the local Hilbert-Samuel function
of I at x = x0 coincides with the local Hilbert-Samuel function of J(x, x0 )
at x = 0.
Applying the arguments of 3.4.1 gives us the answer to our original problem. We notice that Z is empty because in the algorithms above, condition
(2) (i.e. J ⊂ k[x, y]Q) is never satisfied.
STRATIFICATION BY THE LOCAL HILBERT-SAMUEL FUNCTION
17
3.5. A theoritical consequence. Let n, d, q be integers in N. Consider
the variables x = (x1 , . . . , xn ) and a = {aj,α / j = 1, . . . , q; α ∈P
Nn , |α| ≤ d}.
For each j = 1, . . . , q, consider the general polynomial fj = |α|≤d aj,α xα
viewed in Q[a, x], and consider the ideal I = I(x, a) ⊂ Q[x, a] generated by
the fj ’s.
The arguments used above show that for any field K of characteristic 0
there exist a stratification of Kd+n into locally closed subsets satisfying:
• For each strata W , and for any (a0 , x0 ) ∈ W the Hilbert-Samuel
function of I(a0 , x) at x = x0 is constant.
• The stratification is defined by ideals in Q[a, y] and depends only on
the integers d, n and q (in particular it is independent of K).
As a consequence:
• For given d, n, q, the number of possible Hilbert-Samuel functions
(K being arbitrary of characteristic 0) is finite.
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