Bounds for the Hilbert Function of Polynomial Ideals and for the Degrees in the Nullstellensatz

Martín Sombra ^{1}

Departamento de Matemática, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina.

e-mail:  msombra@dm.uba.ar

Abstract. We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the geometric degree of the system of equations. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases.

The proof of this result is combinatorial, and it relies on global estimations for the Hilbert function of homogeneous polynomial ideals.

In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical polynomial ideal.

Introduction

Let k be a field with an algebraic closure denoted by k̄, and let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be polynomials which have no common zero in k̄^{n}. Classical Hilbert’s Nullstellensatz ensures then that there exist polynomials a_{1}, …, a_{s} ∈ k[x_{1}, …, x_{n}] such that

1 = a_{1}f_{1} + … + a_{s}f_{s}

An effective Nullstellensatz amounts to estimate the degrees of the polynomials a_{1}, …, a_{s} in one such a representation. An explicit bound for the degrees reduces the problem of effectively finding the polynomials a_{1}, …, a_{s} to the solving of a system of linear equations.

The effective Nullstellensatz has been the object of much research during the last ten years because of both its theoretical and practical interest. The most precise bound obtained up to now for this problem in terms of the number of variables n and the maximum degree d of the polynomials f_{1}, …, f_{s} is

deg a_{i} ≤ max{3, d}^{n} 1 ≤ i ≤ s

This bound is due to J. Kollar [22], and it is essentially optimal for d ≥ 3; in the case when d = 2 a sharper estimate can be given (see [30]).

1Partially supported by CONICET PID 3949/92 and UBA–CYT EX. 001.

Related results can be found in the research papers [7], [8], [12], [31], [29], [4], [2], [23], also there are extensive discussions and bibliography about the effective Nullstellensatz in the surveys [33], [3].

Because of its exponential nature, this bound is hopeless for most practical applications. This behavior is in general unavoidable for polynomial elimination problems when only the number of variables and the degrees of the input polynomials are considered.

However, it has been observed that there are many particular cases in which this bound can be notably improved. This fact has motivated the introduction of new parameters which enable to differenciate special families of systems of polynomial equations whose behavior for the problem in question is, say, polynomial instead of exponential (see [15], [14]).

In this spirit, we consider an additional parameter associated to the input polynomials f_{1}, …, f_{s}, called the geometric degree of the system of equations, which is defined as follows.

Let k be a zero characteristic field and f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] polynomials such that 1 ∈ (f_{1}, …, f_{s}). Then there exist t ≤ s and g_{1}, …, g_{t} k̄–linear combinations of f_{1}, …, f_{s} such that 1 ∈ (g_{1}, …, g_{t}), g_{1}, …, g_{t−1} is a regular sequence, and (g_{1}, …, g_{t−1}) ⊆ k[x_{1}, …, x_{n}] is a radical ideal for 1 ≤ i ≤ t − 1. Let V_{i} ⊆ IA^{n}(k̄) be the affine variety defined by g_{1}, …, g_{i} for 1 ≤ i ≤ s, and set

δ_{g_{1},...,g_{s}} := max_{1≤i≤min{t,n}−1} deg V_{i}

where deg V_{i} stands for the degree of the affine variety V_{i}. Then the geometric degree of the system of equations δ(f_{1}, …, f_{s}) is defined as the minimum of the δ_{g_{1},...,g_{s}} through all linear combinations of f_{1}, …, f_{s} satisfying the stated conditions.

In the case when k is a field of positive characteristic, the degree of the system of equations f_{1}, …, f_{s} is defined in an analogous way by considering k̄–linear combinations of the polynomials {f_{i}, x_{j}f_{i} : 1 ≤ i ≤ s, 1 ≤ j ≤ n}.

In both cases, the existence of g_{1}, …, g_{s} satisfying these properties is a consequence of Bertini’s theorem.

We obtain (Theorem 4.40):

Theorem. Let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be polynomials such that 1 ∈ (f_{1}, …, f_{s}). Let d := max_{1≤i≤s} deg f_{i}, and let δ be the geometric degree of the system of equations f_{1}, …, f_{s}. Then there exist a_{1}, …, a_{s} ∈ k[x_{1}, …, x_{n}] such that

1 = a_{1}f_{1} + … + a_{s}f_{s}

with deg a_{i}f_{i} ≤ min{n, s}^{2} (d + 3n) δ for i = 1, …, s.

We also obtain a similar bound for the representation problem in complete intersections (Theorem 4.39).

Let d := max_{1≤i≤s} deg f_{i}. Then we have

δ(f_{1}, …, f_{s}) ≤ (d + 1)^{min{s,n}−1}

and so our bounds for the effective Nullstellensatz and for the representation problem in complete intersections are essentially sharp in the general case. We remark however that they can substantially improve the usual estimates in some special cases (see Example 4.42).

Similar bounds for the effective Nullstellensatz have also been recently obtained by algorithmic tools [15, Th. 19], [14, §4.2] and by duality methods [24].

The proof of these theorems are combinatorial, and rely on global estimations for the Hilbert function of certain polynomials ideals.

The study of the global behavior of the Hilbert function of homogeneous ideals is of independent interest. It is related to several questions of effective commutative algebra, mainly in connexion with the construction of regular sequences of maximal length with polynomials of controlled degree lying in a given ideal [10, §2], and to trascendental number theory, in the context of the so–called zero lemmas [5].

Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal. We understand for the dimension of I the dimension of the projective variety that it defines, and we denote by h_{I} its Hilbert function.

The problem of estimating h_{I} was first considered by Y. Nesterenko [28], who proved that for a zero characteristic field k and an homogeneous prime ideal P ⊆ k[x_{0}, …, x_{n}] of dimension d ≥ 0 it holds

(_{d+1}^{m+d+1}) − (_{d+1}^{m−deg P +d+1}) ≤ h_{P} (m) ≤ deg P (4 m)^{d} m ≥ 1

Later on, M. Chardin [10] improved Nesterenko’s upper bound by simplifying his proof, and obtained that for a perfect field k and an homogeneous unmixed radical ideal I ⊆ k[x_{0}, …, x_{n}] of dimension d ≥ 0 it holds

h_{I}(m) ≤ deg I (_{d}^{m+d}) m ≥ 1

This estimation has also been obtained by J. Kollar (see [10, Note]) by cohomological arguments.

In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal of dimension d ≥ 0 (Theorem 2.4). We have:

h_{I}(m) ≥ (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg I+d+1}) m ≥ 1

This result generalizes the bound of Y. Nesterenko [28] for the case of an homogeneous prime ideal P ⊆ k[x_{0}, …, x_{n}]. It is optimal in terms of the dimension and the degree of the ideal I.

We present also an upper bound for the Hilbert function of a generic hypersurface section f of an homogeneous unmixed radical ideal I ⊆ k[x_{0}, …, x_{n}] of dimension d ≥ 1 (Theorem 2.23). We have:

h_{(I,f)}(m) ≤ 3 deg f deg I (_{d−1}^{m+d−1})

for m ≥ 5 d deg I.

Our approach to the Hilbert function is elementary, and yields a new point of view into the subject which is clearer than that of the previous works. We hope that our techniques would also be useful for treating arithmetic Hilbert functions (see [28]).

We shall briefly sketch the relationship between these bounds for the Hilbert function, and the effective Nullstellensatz and the representation problem in complete intersections.

Let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be a regular sequence. There are several effectivity questions about this set of polynomials which can be easily solved in the case when the homogenization of these polynomials f̃_{1}, …, f̃_{s} ∈ k[x_{0}, …, x_{n}] is again a regular sequence. An example of this situation is the effective Nullstellensatz, for which there exists a simple and well–known proof in this conditions (see for instance [26, §1, Th. 1, Cor.]).

The central point in our proof of the effective Nullstellensatz consists then in showing that the regular sequence f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] can in fact be replaced by polynomials p_{1}, …, p_{s} ∈ k[x_{1}, …, x_{n}] of controlled degrees such that (f_{1}, …, f_{i}) = (p_{1}, …, p_{i}) for 1 ≤ i ≤ s, and such that the homogenizated polynomials p̃_{1}, …, p̃_{s} define a regular sequence in k[x_{0}, …, x_{n}]. The proof of this result proceeds by induction, and the bounds for the Hilbert function allows us to controlle at each step 1 ≤ i ≤ s the degree of the polynomial p_{i}.

The spirit of our proof follows T. Dubé’s paper on the effective Nullstellensatz [11]. We remark here that there are many errors in Dubé’s argument, and a serious gap, for it relies on an assumption on the Hilbert function of certain class of homogeneous polynomial ideals [11, §2.1] which is unproved in his paper and which is neither in the literature, as it was noted by M. Almeida [1, §3.1], and thus this proof should be considered incomplete as it stands.

Our approach allows us not only to avoid Dubé’s assumption and prove the results stated in his paper, but also to obtain our more refined bounds.

Finally, we want to remark that our exposition is elementary and essentially self–contained.

The exposition is divided in four parts. In the first we state some well–known features of degree of projective varieties and Hilbert function that will be needed in the subsequent parts, and prove some of them when suitable reference is lacking. In the second part we prove the lower and upper bounds for the Hilbert function and analize the extremal cases. In the third part, we apply the obtained results to the construction of regular sequences. In the fourth part we consider the consequences for the effective Nullstellensatz and for the representation problem in complete intersections.

I am specially grateful to P. Solernó for many discussions and suggestions which substantially improved the paper. I also want to acknowledge J. Heintz for many helpful suggestions and encouragement, T. Krick for her assistance during the redaction of the paper, L. M. Pardo and G. Valla for their comments on both the contents and the presentation of the paper, P. Philippon for pointing me out an error in a previous version of this paper, and M. Almeida, who turned my attention to T. Dubé’s work on the effective Nullstellensatz.

§ 0. Notations and Conventions

We work over an arbitrary field k with algebraic closure k̄. As usual, IP^{n} and IA^{n} denote the projective space and the affine space of dimension n over k̄. A variety is not necessarily irreducible.

The ring k[x_{0}, …, x_{n}] will be denoted alternatively by R or R_{k}.

Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal. We understand for the dimension of I the dimension of the projective variety that it defines and we shall denote it by dim I, so that dim I = dimkrull I − 1.

Let J ⊆ k[x_{1}, …, x_{n}] be an affine ideal. We shall understand for the dimension of J its Krull dimension. In each appereance, it will be clear from the context to which notion we are refering to.

An ideal I ⊆ k[x_{0}, …, x_{n}] is unmixed if its associated prime ideals have all the same dimension. In particular, I has not imbedded associated primes and its primary decomposition is unique.

Given an ideal I ⊆ k[x_{0}, …, x_{n}], then I^{e} := k̄⊗_{k}I ⊆ k̄[x_{0}, …, x_{n}] is the extended ideal of I in k̄[x_{0}, …, x_{n}].

Given I ⊆ R_{k} an homogeneous ideal,

V (I) := {x ∈ IP^{n} : f(x) = 0 ∀f ∈ I} ⊆ IP^{n}

is the projective variety defined by I. Conversely, given V ⊆ IP^{n} a variety,

I_{k}(V) := {f ∈ R_{k} : f|_{V} ≡ 0} ⊆ R_{k}

and I(V) := Ik̄(V) ⊆ k̄[x_{0}, …, x_{n}] is the defining ideal of V.

Given a graded R–module M and m ∈ ZZ, M_{m} is the homogeneous part of degree m.

Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal. The Hilbert function or characteristic function of the ideal I is

\begin{aligned}h_{I} : & ZZ → ZZ \\ & m ↦ dim_{k}(k[x_{0}, …, x_{n}]/I)_{m} \end{aligned}

Given V ⊆ IP^{n} a variety, then h_{V} is the Hilbert function of I(V).

Given f ∈ k[x_{0}, …, x_{n}] an homogeneous polynomial, f^{a} ∈ k[x_{1}, …, x_{n}] is its affinization and given I ⊆ k[x_{0}, …, x_{n}] an homogeneous ideal, I^{a} ⊆ k[x_{1}, …, x_{n}] is its affinization.

Conversely, given an affine polynomial g ∈ k[x_{1}, …, x_{n}], g̃ ∈ k[x_{0}, …, x_{n}] is its homogenization, and given J ⊆ k[x_{1}, …, x_{n}] an affine ideal, we denote by J̃ ⊆ k[x_{0}, …, x_{n}] its homogenization.

§ 1. Preliminaries on Degree and Hilbert Function

In this section we state some well–known properties concerning the degree of a variety and the Hilbert function of an homogeneous polynomial ideal which will be needed in the sequel. Also we shall prove some of them when suitable reference is lacking.

Let V ⊆ IP^{n} be an irreducible projective variety of dimension d. The degree of V is defined as

deg V := sup { #(V ∩ H_{1} ∩ … ∩ H_{d}) : H_{1}, …, H_{d} ⊆ IP^{n} hyperplanes
and dim(V ∩ H_{1} ∩ … ∩ H_{d}) = 0}

This number is finite, and it realizes generically, if we think the set

{(H_{1}, …, H_{d}) : H_{1}, …, H_{d} ⊆ IP^{n} hyperplanes }

as parametrized by a non empty set of IA^{(n+1)d} [17, Lecture 18]. We agree that deg ∅ = 1.

The notion of degree can be extended to possible reducible projective varieties following [19]. Let V ⊆ IP^{n}, and let V = ∪ C be the minimal decomposition of V in irreducible varieties. Then the (geometric) degree of V is defined as

deg V := ∑_{C} deg C

For this notion of degree it holds the following Bézout’s inequality without multiplicities for the degree of the intersection of two varieties. Let V, W ⊆ IP^{n} be varieties. Then we have

deg(V ∩ W) ≤ deg V deg W

This is a consequence of Bézout’s inequality for affine varieties [19, Th. 1]. The details can be found in [9, Prop. 1.11]. This result can also be deduced from the Bézout’s theorems [13, Th. 12.3], [34, Th. 2.1].

Also the notion of degree and the Bézout’s theorem traslate to the affine context.

We turn our attention to the Hilbert function of an homogeneous ideal. Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal of dimension d. There exists a polynomial p_{I} ∈ Q[t] of degree d, and m_{0} ∈ ZZ such that

h_{I}(m) = p_{I}(m)

for m ≥ m_{0}. The polynomial p_{I} is called the Hilbert polynomial of the ideal I.

The degree of an homogeneous ideal I ⊆ k[x_{0}, …, x_{n}] can be defined through its Hilbert polynomial.

Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal of dimension d, with d ≥ 0. Let p_{I} = a_{d} t^{d} + … + a_{0} ∈ Q[t] be its Hilbert polynomial. Then the (algebraic) degree of the ideal I is defined as

deg I := d! a_{d} ∈ IN

If I ⊆ k[x_{0}, …, x_{n}] is an homogeneous ideal of dimension –1, then I is a (x_{0}, …, x_{n})–primary ideal, and the degree of I is defined as the length of the k–module k[x_{0}, …, x_{n}]/I, which equals its dimension as a k–linear space. We also agree that deg k[x_{0}, …, x_{n}] = 0.

Given I ⊆ k[x_{0}, …, x_{n}] an homogeneous ideal, we denote by irr I the number of irreducible components of V (I) ⊆ IP^{n}.

Let I, J ⊆ k[x_{0}, …, x_{n}] be homogeneous ideals. Then we have the following exact sequence of graded k–algebras

\begin{aligned}0 → R/I ∩ J → & R/I ⊕ R/J & → & R/I + J & → 0 \\ & (f, g) & ⟼ & f − g \end{aligned}

from where we get

h_{I∩J}(m) = h_{I}(m) + h_{J}(m) − h_{I+J}(m) m ≥ 1

In particular, if dim I > dim J, then deg I ∩ J = deg I.

Let k be a perfect field, I ⊆ k[x_{0}, …, x_{n}] an homogeneous radical ideal, and let I = ∩_{P} P be the minimal primary decomposition of I. In this situation we have

deg I = ∑_{P: dim P =dim I} deg V (P)

(see [34, Prop. 1.49] [17, Prop. 13.6]), and thus the degree of the ideal I may be calculated from the degrees of the varieties defined by its associated prime ideals of maximal dimension.

Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous radical ideal, and I = ∩_{P} P the minimal primary decomposition of I. From the canonical inclusion of graded modules

R_{k}/I ↪ ⨁_{P} R_{k}/P

we deduce that

h_{I}(m) ≤ ∑_{P} h_{P} (m) m ≥ 1

Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal, and I^{e} ⊆ k̄[x_{0}, …, x_{n}] be the extended ideal. Let

k[x_{0}, …, x_{n}]/I = ⨁_{m}(k[x_{0}, …, x_{n}]/I)_{m}

be the decomposition of k[x_{0}, …, x_{n}]/I into homogeneous parts. Then

(k̄[x_{0}, …, x_{n}]/I^{e})_{m} = k̄ ⊗_{k} (k[x_{0}, …, x_{n}]/I)_{m} m ∈ ZZ

and so h_{I^{e}}(m) = h_{I}(m), i.e. the Hilbert function is invariant under change of the base field. In particular

deg I^{e} = deg I

We have also that there exist y_{0}, …, y_{d} ∈ k̄[x_{0}, …, x_{n}] algebraically independent linear forms such that

k̄[y_{0}, …, y_{d}] ↪ k̄[x_{0}, …, x_{n}]/I^{e}

is an inclusion of k̄–algebras, and so

h_{I}(m) = h_{I^{e}}(m) ≥ dimk̄(k̄[y_{0}, …, y_{d}])_{m} = (_{d}^{m+d})

We shall need the following identity for the combinatorial numbers.

Lemma 1.1 Let d ≥ 0, D ≥ 1, m ∈ ZZ. Then

(_{d+1}^{m+d+1+D}) − (_{d+1}^{m+d+1}) = ∑_{i=1}^{D} (_{d}^{m+d+i})

Proof. The case D = 1 is easy. In the case when D > 1, we have

(_{d+1}^{m+d+1+D}) − (_{d+1}^{m+d+1}) = ∑_{i=1}^{D}{(_{d+1}^{m+d+i+1}) − (_{i}^{m+d+i})} = ∑_{i=1}^{D} (_{d}^{m+d+i})

□

We shall make appeal also to Macaulay’s characterization of the Hilbert function of an homogeneous polinomial ideal.

Given positive integers i, c, the i–binomial expansion of r is the unique expression

c = (_{i}^{c(i)}) + … + (_{j}^{c(j)})

with c(i) > … > c(j) ≥ j ≥ 1.

Let c = (_{i}^{c(i)}) + … + (_{j}^{c(j)}) be the i–binomial expansion of c. Then we set

c^{⟨i⟩} := (_{i+1}^{c(i)+1}) + … + (_{j+1}^{c(j)+1})

We note that this expression is the i + 1–binomial expansion of c^{⟨i⟩}.

Remark 1.2 Let b, c, i ∈ ZZ_{>0}. Then it is easily seen that b ≥ c if and only if (b(i), …, b(j)) is greater or equal that (c(i), …, c(j)) in the lexicographic order, and thus and thus b ≥ c if and only if b^{⟨i⟩} ≥ c^{⟨i⟩}.

We recall that a sequence of nonnegative integers (c_{i})_{i∈ZZ_{≥0}} is called an O– sequence if

c_{0} = 1  c_{i+1} ≤ c_{i}^{⟨i⟩}  i ≥ 1

We have then

Theorem(Macaulay, [16]). Let h : ZZ_{≥0} → ZZ_{≥0}. Then h is the Hilbert function of an homogeneous polynomial ideal if and only if

(h(i))_{i∈ZZ_{≥0}}

is an O–sequence.

□

§ 2. Bounds for the Hilbert Function

In this section we shall derive both lower and upper bounds for the Hilbert function of homogeneous polynomial ideals. These estimations depend on the dimension and on the degree of the ideal in question, and eventually on its length.

We derive first a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal.

We shall consider separately the case when dim I = 0.

Lemma 2.3 Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous unmixed ideal of dimension 0. Then

h_{I}(m) ≥ m + 1  deg I − 2 ≥ m ≥ 0
h_{I}(m) = deg I  m ≥ deg I − 1

Proof. We have that I^{e} ⊆ k̄[x_{0}, …, x_{n}] is an unmixed ideal of dimension 0 [35, Ch. VII, §31, Th. 36, Cor.1]. As k̄ is an infinite field, there exist a linear form u ∈ k̄[x_{0}, …, x_{n}] which is a non zero divisor modulo I^{e}. Then

h_{I}(m) − h_{I}(m − 1) = h_{I^{e}}(m) − h_{I^{e}}(m − 1) = h_{(I^{e},u)}(m)

Let m_{0} be minimum such that

h_{I^{e}}(m) = deg I^{e} = deg I

for m ≥ m_{0}. Then h_{(I^{e},u)}(m) ≥ 1 for 0 ≤ m ≤ m_{0} − 1 and h_{(I^{e},u)}(m) = 0 for m ≥ m_{0}, and thus we have

h_{I}(m) = h_{I^{e}}(m) ≥ m + 1 deg I − 2 ≥ m ≥ 0

and also h_{I}(m) = h_{I^{e}}(m) = deg I for m ≥ deg I − 1. □

Theorem 2.4 Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal of dimension d, with d ≥ 0. Then

h_{I}(m) ≥ (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg I+d+1}) m ≥ 1

Proof. Let I^{e} = ⋂_{P} Q_{P} be a minimal primary decomposition of I^{e}, and let

I^{∗} = ∩_{dim P =dim I^{e}} Q_{P}

be the intersection of the primary components of I^{e} of maximal dimension, which is an unmixed ideal of dimension d. Then h_{I}(m) = h_{I^{e}}(m) ≥ h_{I^{∗}}(m) for m ≥ 1, and we have that

deg I = deg I^{∗}

We shall proceed by induction on d. Consider first d = 0. We have then

h_{I}(m) = h_{I^{e}}(m) ≥ h_{I^{∗}}(m) ≥ (_{1}^{m+1}) − (_{1}^{m−deg I+1}) m ≥ 1

by Lemma 2.3 applied to I^{∗}.

Now let d ≥ 1. Let u ∈ k̄[x_{0}, …, x_{n}] be a linear form which is a non zero divisor modulo I^{∗}. Then we have

h_{I^{∗}}(m) − h_{I^{∗}}(m − 1) = h_{(I^{∗},u)}(m)

Then dim(I, u) = d − 1 and deg (I^{∗}, u) = deg I^{∗} = deg I. By the inductive hypothesis we have that

h_{I^{∗}}(m) − h_{I^{∗}}(m − 1) = h_{(I^{∗},u)}(m) ≥ (_{d}^{m+d}) − (_{d}^{m−deg I+d}) m ≥ 1

Then

\begin{aligned}h_{I}(m) & ≥ & h_{I^{∗}}(m) = ∑_{j=0}^{m} h_{(I^{∗},u)}(j) ≥ \\ & ≥ & ∑_{j=0}^{m}\{(_{d}^{j+d}) − (_{d}^{j−deg I+d})\} = (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg I+d+1}) m ≥ 1 \end{aligned}

by Lemma 1.1. □

This inequality extends Nesterenko’s estimate for the case of a prime ideal [28, §6, Prop. 1] to the case of an arbitrary ideal.

Remark 2.5 By Gotzmann’s persistence theorem [16] we have that for an homogeneous ideal I ⊆ k[x_{0}, …, x_{n}] of dimension d there exists m_{0} ∈ ZZ such that

h_{I}(m) ≥ (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg I+d+1}) m ≥ m_{0}

as it is noted in [6, Rem. 0.6]. Our theorem shows that this inequality holds globally, not only for big values of m.

Given I ⊆ k[x_{0}, …, x_{n}] an homogeneous ideal of dimension d ≥ 0, let

H_{I}(t) := ∑_{m=0}^{∞} h_{I}(m) t^{m}

be its Hilbert–Poincaré series. Then the previous result states that

H_{I}(t) ≥ \frac{1 − t^{deg I}}{(1 − t)^{d+2}}

in the sense that the inequality holds for each term of the power series.

This estimate is optimal in terms of the dimension and the degree of the ideal I. The extremal cases correspond to hypersurfaces of linear subspaces of IP^{n}. This can be deduced from [6, Cor. 2.8], which in turn depends on Gotzmann’s theorem. We give here a self–contained proof of this fact.

Proposition 2.6 Let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal. Then

h_{I}(m) = (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg I+d+1}) m ≥ 1

if and only if there exist u_{1}, …, u_{n−d−1} ∈ k[x_{0}, …, x_{n}] linearly independent linear forms and f ∉ (u_{1}, …, u_{n−d−1}) such that I = (u_{1}, …, u_{n−d−1}, f).

Proof. Let u_{1}, …, u_{n−d−1} ∈ k[x_{0}, …, x_{n}] be linearly independent linear forms and f ∉ (u_{1}, …, u_{n−d−1}). Let I := (u_{1}, …, u_{n−d−1}, f). Then f is a non zero divisor modulo (u_{1}, …, u_{n−d−1}), and so

h_{I}(m) = h_{(u_{1},...,u_{n−d−1})}(m) − h_{(u_{1},...,u_{n−d−1})}(m − deg f) =
= (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg f+d+1}) = (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg I+d+1})

Conversely, let I ⊆ k[x_{0}, …, x_{n}] be an homogeneous ideal such that

h_{I}(m) = (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg I+d+1}) m ≥ 1

Then h_{I}(1) ≤ d + 2, i. e. dim_{k} I_{1} ≥ n − d − 1. Let u_{1}, …, u_{n−d−1} ∈ I_{1} be linearly independent linear forms. We have

h_{I}(m) = (_{d+1}^{m+d+1})  deg I − 1 ≥ m ≥ 1
h_{I}(deg I) = (_{d+1}^{deg I+d+1}) − 1

Thus

h_{I}(m) = h_{(u_{1},...,u_{n−d−1})}(m)  deg I − 1 ≥ m ≥ 1
h_{I}(deg I) < h_{(u_{1},...,u_{n−d−1})}(deg I)

and so there exist f ∈ I − (u_{1}, …, u_{n−d−1}) with deg f = deg I. Let J := (u_{1}, …, u_{n−d−1}, f) ⊆ k[x_{0}, …, x_{n}]. Then J ⊆ I and h_{J}(m) = h_{I}(m) for all m ≥ 0, and thus we have J = I. □

We devote now to the upper bounds. In this respect we have two different estimates. The first bound is sharp for small values and the second for big ones.

The first upper bound will be deduced from a series of results and observations.

Definition 2.7 Let V ⊆ IP^{n} be a variety. Then the linear closure of V is the smallest linear subspace of IP^{n} which contains V, and it is denoted by L(V).

Remark 2.8 Let E ⊆ IP^{n} be a linear space. Then its defining ideal I(E) ⊆ Rk̄ is generated by linear forms, and it is easy to see that

dim E = n − dimk̄ I(E)

Let V ⊆ IP^{n} be a variety, and let L ∈ Rk̄ linear form. Then L|_{V} ≡ 0 if and only if L|_{L(V)} ≡ 0, and thus

I(L(V)) = (I(V)_{1}) ⊆ k̄[x_{0}, …, x_{n}]

In particular we have

h_{V} (1) = n + 1 − dimk̄ Ik̄(V)_{1} = dim L(V) + 1

The following proposition shows that the dimension of the linear closure is bounded in terms of the dimension and the degree of the variety. It is a consequence of Bertini’s theorem [21, Th. 6.3]. A proof can be found in [17, Cor. 18.12].

Proposition 2.9 Let V ⊆ IP^{n} be an irreducible variety. Then

dim L(V) + 1 ≤ deg V + dim V
□

The following is an estimation of the degree of the image of a variety under a regular map. It is a variant of [20, Lemma 1] and [30, Prop. 1].

Proposition 2.10 Let V ⊆ IP^{n} be a variety, f_{0}, …, f_{N} ∈ k̄[x_{0}, …, x_{n}] homogeneous polynomials of degree D which define a regular map

\begin{aligned}φ : & IP^{n} ⟶ IP^{N} \\ & x := (x_{0} : …: x_{n}) ↦ (f_{0}(x) : …: f_{N}(x)) \end{aligned}

Then deg φ(V) ≤ deg V D ^{dim V}.

Proof. We can suppose without loss of generality that V is irreducible. Let d := dim φ(V), and let H_{1}, …, H_{d} ⊆ IP^{n} be hyperplanes such that

#(φ(V) ∩ H_{1} ∩ … ∩ H_{d}) = deg φ(V)

For i = 1, …, d, let L_{i} ∈ Rk̄ be linear forms such that H_{i} = {L_{i} = 0}. Then

#(φ(V) ∩ H_{1} ∩ … ∩ H_{d})

is bounded by the number of irreducible components of φ^{−1}(φ(V) ∩ H_{1} ∩ … ∩ H_{d}) and so we have

#(φ(V) ∩ H_{1} ∩ … ∩ H_{d}) ≤ deg φ^{−1}(φ(V) ∩ H_{1} ∩ … ∩ H_{d}) =
= deg (V ∩ ⋂_{i=1}^{d} V (L_{i}(f_{0}, …, f_{N}))) ≤ deg V D^{d}

by Bézout’s inequality. We have then

deg φ(V) ≤ deg V D ^{dim V}

as dim φ(V) ≤ dim V. □

Now it follows easily the desired inequality for the case of an irreducible variety.

Proposition 2.11 Let V ⊆ IP^{n} be an irreducible variety of dimension d, with d ≥ 0. Then

h_{V} (m) ≤ deg V m^{d} + d m ≥ 1

Proof. For n, m ∈ IN, let

\begin{aligned}v_{m} : & IP^{n} ⟶ IP^{(_{n}^{n+m})} \\ & (x_{0} : …: x_{n}) ↦ (x^{(i)})_{|i|=m} \end{aligned}

be the Veronese map of degree m. Then v_{m}|_{V} : V ↦ v_{m}(V) is a birregular morphism of degree m, and so we have that

h_{v_{m}(V)} (k) = h_{V} (mk) k ≥ 1

In particular we have that

h_{V} (m) = h_{v_{m}(V)} (1) = dim L(V) + 1

by Remark 2.8, and so

h_{V} (m) ≤ deg v_{m}(V) + dim v_{m}(V) ≤ deg V m^{d} + d

by application of Propositions 2.9 and 2.10. □

We can extend this bound to the more general case of an unmixed radical ideal in k[x_{0}, …, x_{n}].

Theorem 2.12 Let k be a perfect field and I ⊆ k[x_{0}, …, x_{n}] an homogeneous unmixed radical ideal of dimension d, with d ≥ 0. Then

h_{I}(m) ≤ deg I m^{d} + irr I d m ≥ 1

Proof. Let I^{e} ⊆ Rk̄ be the extended ideal of I in Rk̄. Then I^{e} is an unmixed radical ideal of dimension d [35, Ch. VII, §31, Th. 36, Cor. 1] [27, Th. 26.3]. Let I^{e} = ∩_{P} P be the minimal primary decomposition of I^{e}. Then we have

h_{I}(m) ≤ ∑_{P} h_{P} (m)

from where

h_{I}(m) ≤ ∑_{P} (degV (P) m^{d} + d) = deg I m^{d} + irr I d m ≥ 1

by Proposition 2.11. □

This inequality has the same order of growth of h_{I}. We see also that it does not improve the estimate

h_{I}(m) ≤ deg I (_{d}^{m+d−1}) + irr I (_{d−1}^{m+d−1}) m ≥ 1

which follows from Chardin’s arguments [10].

¿From the asymptotic behavior h_{I}(m) ∼ \frac{deg I}{d!} m^{d} we see that this inequality is sharp for big values of m only when d = 1. In this case, the inequality is optimal in terms of the degree and the length of the ideal, and we determine the extremal cases.

Definition 2.13 Let V, W ⊆ IP^{n} be varieties. Then V, W are projectively equivalent if there exist an automorphism A ∈ P GL_{n+1}(k̄) such that W = A(V) [17, p. 22].

Remark 2.14 Let V, W ⊆ IP^{n} be varieties. Then V, W are projectively equivalent if and only if its coordinated rings k̄[V], k̄[W] are isomorphic as graded k̄–algebras. In particular their Hilbert function coincide.

A curve C ⊆ IP^{n} is called a rational normal curve if it is projectively equivalent to v_{n}(IP^{1}). Then C is non degenerated, i. e. L(C) = IP^{n} [17, Example 1.14], and its degree is n [17, Exerc. 18.8]. By Proposition 2.9 the degree of C is minimum with the condition of being non degenerated. In fact, rational normal curves are characterized by these two properties [17, Prop. 18.9].

Now let l, n ∈ IN, δ = (δ_{1}, …, δ_{l}) ∈ IN^{l} such that |δ| := δ_{1} + … + δ_{l} ≤ n + 1 − l. For 1 ≤ j ≤ l, let n_{j} := δ_{1} + … + δ_{j} + j, and consider the inclusion of linear spaces given by

\begin{aligned}i_{j} : & IP^{δ_{j}} ↪ IP^{n} \\ & (x_{0} : …: x_{δ_{j}}) ↦ (\overbrace{0 : …: 0}^{n_{j−1}} : x_{0} : …: x_{δ_{j}} : 0 : …: 0) \end{aligned}

The linear subspaces i_{j}(IP^{δ_{j}}) ⊆ IP^{n} are disjoint one from each other. Let

C(n, δ) := ⋃_{j=1}^{l} i_{j}(v_{δ_{j}}(IP^{1})) ⊆ IP^{n}

A curve C ⊆ IP^{n} is projectively equivalent to C(n, δ) if and only if there exist E_{1}, …, E_{l} ⊆ IP^{n} disjoint linear subspaces such that dim E_{j} = δ_{j}, C ⊆ ∪_{j}E_{j}, and

C_{j} := C ∩ E_{j} ⊆ E_{j}

is a rational normal curve for 1 ≤ j ≤ l.

Definition 2.15 Let V ⊆ IP^{n} be a variety. Then V is defined over k if Ik̄(V) = k̄ ⊗_{k} I_{k}(V) ⊆ k̄[x_{0}, …, x_{n}], i. e. if its defining ideal is generated over k.

The following lemma is well–known, we prove it here for lack of suitable reference.

Lemma 2.16 Let φ : IP^{n} → IP^{N} be a regular map defined over k, V ⊆ IP^{n} be a variety defined over k. Then φ(V) ⊆ IP^{N} is defined over k.

Proof. We have the following conmutative diagram

k[x_{0}, …, x_{N}] ⟶^{φ_{k}^{∗}} k[V]
↓  ↓
k̄[x_{0}, …, x_{N}] ⟶^{φ∗ ; k̄} k̄[V]

with ker φ_{k}^{∗} = I_{k}(W) and ker φ∗ ; k̄ = Ik̄(W). We have k̄ ⊗_{k} k[V] ≅ k̄[V] as V is defined over k, and tensoring with k̄ we get

k̄[x_{0}, …, x_{N}] ⟶^{k̄⊗_{k}φ_{k}^{∗}} k̄ ⊗_{k} k[V]
∥  ∥
k̄[x_{0}, …, x_{N}] ⟶^{φ∗ ; k̄}  k̄[V]

with ker k̄ ⊗_{k} φ_{k}^{∗} = k̄ ⊗_{k} I_{k}(W), from where we deduce Ik̄(W) = k̄ ⊗_{k} I_{k}(W), i. e. Ik̄(W) is defined over k. □

Let v_{n} : IP^{1} → IP^{n} be the Veronese map of degree n and let C_{n} := v_{n}(IP^{1}) be its image. By the preceeding lemma, C_{n} is defined over k.

For 1 ≤ j ≤ l, let C_{j} := i_{j}(C_{δ_{j}}) ⊆ IP^{n}. Then

C(n, δ) = ⋃_{j=1}^{l} i_{j}(v_{δ_{j}}(IP^{1}))

is the minimal decomposition of C(n, δ) in irreducible curves. Thus C(n, δ) is also defined over k, and so

irr I_{k}(C(n, δ)) = l, deg I_{k}(C(n, δ)) = |δ|

Lemma 2.17 Let V, W ⊆ IP^{n} be varieties. Then

I(V) + I(W) = (x_{0}, …, x_{n})

if and only if V, W lie in disjoint linear subspaces of IP^{n}.

Proof. Given V, W ⊆ IP^{n} varieties, they lie in disjoint linear subspaces if and only if

L(V) ∩ L(W) = ∅

Let L_{V} := I(L(V)), L_{W} := I(L(W)). By Remark 2.8 we have

\begin{matrix}L_{V} = (I(V)_{1}) ⊆ I(V) \\ L_{W} = (I(W)_{1}) ⊆ I(W) \end{matrix}

In particular L_{V}, L_{W} are generated by linear forms, and so

L_{V} + L_{W} = I(L(V) ∩ L(W))

Let V, W ⊆ IP^{n} such that L(V) ∩ L(W) = ∅. Then

L_{V} + L_{W} = (x_{0}, …, x_{n})

and so I(V) + I(W) = (x_{0}, …, x_{n}). Conversely, suppose that I(V) + I(W) = (x_{0}, …, x_{n}). Then

x_{0}, …, x_{n} ∈ I(V)_{1} + I(W)_{1}

Thus L_{V} + L_{W} = (x_{0}, …, x_{n}) and so L(V) ∩ L(W) = ∅ □

Proposition 2.18 Let k be a perfect field and I ⊆ k[x_{0}, …, x_{n}] an homogeneous unmixed radical ideal of dimension 1. Then

h_{I}(m) = deg I m + irr I m ≥ 1

if and only if there exist δ ∈ IN^{l} with l := irr I, such that |δ| = deg I, and a curve C ⊆ IP^{n} defined over k projectively equivalent to C(n, δ) such that I = I_{k}(C).

Proof. Let C ⊆ IP^{n} be a curve defined over k projectively equivalent to C(n, δ) for some δ ∈ IN^{l} and l = irr I. Then k̄ ⊗_{k} I_{k}(C) = Ik̄ ⊆ Rk̄ and so

irr I_{k}(C) = irr I(C(n, δ)) = l, deg I_{k}(C) = deg I(C(n, δ)) = |δ|

We aim at proving that

h_{I_{k}(C)}(m) = |δ| m + l m ≥ 1

We have that h_{I_{k}(C)}(m) = h_{C}(m) = h_{C(n,δ)}(m) and so it suffices to prove that

h_{C(n,δ)}(m) = |δ|m + l m ≥ 1

We shall proceed by induction on l. Let C_{d} := v_{d}(IP^{1}). We have the inclusion of graded k̄–algebras

\begin{aligned}k̄[C_{d}] ≅ & k̄[x_{0}, …, x_{d}]/Ik̄(C_{d}) ↪^{v_{d}^{∗}} k̄[x, y] \\ & x_{i} ⟼ x^{i}y^{d−i} \end{aligned}

We have then

k̄[C_{d}] ≅ ⨁_{j=0}^{∞} k̄[x, y]_{d j}

from where h_{C_{d}}(m) = d m + 1 for m ≥ 1, and so the assertion holds for l = 1. Let l > 1, and let

C(n, δ) = ⋃_{j} C_{j}

be the minimal decomposition of C(n, δ) in irreducible curves. Then C_{1} ∪ … ∪ C_{l−1}, C_{l} lie in disjoint linear spaces and so

I(C_{1} ∪ … ∪ C_{l−1}) + I(C_{l}) = (x_{0}, …, x_{n})

by Lemma 2.17. We have then

h_{C}(m) = h_{C_{1}∪...∪C_{l−1}}(m) + h_{C_{l}}(m) m ≥ 1

and from the inductive hypothesis we get

h_{C}(m) = {(δ_{1} +…+δ_{l−1}) m+(l −1)}+{δ_{l} m+1} = |δ| m+l m ≥ 1

Now we shall prove the converse. We have that I^{e} is a radical ideal, and so I^{e} is the ideal of some curve C ⊆ IP^{n} defined over k.

We shall proceed by induction on l := irr I. Let l = 1, i. e. C ⊆ IP^{n} irreducible. Then

dim L(C) = h_{C}(1) − 1 = deg C

and so C ⊆ L(C) is a non degenerated irreducible curve of minimal degree. We have then that C ⊆ L(C) is a rational normal curve [17, Prop. 18.9].

Let l > 1, and suppose that the assertion is proved for l(I) ≤ l − 1 and K an arbitrary field. In particular it is proved for k̄, the algebraic closure of k. Let C = C_{1} ∪ … ∪ C_{l} be the minimal decomposition of C in irreducible curves. Then

h_{C}(m) = h_{C_{1}∪...∪C_{l−1}}(m) + h_{C_{l}}(m) − h_{I(C_{1}∪...∪C_{l−1})+I(C_{l})}(m) m ≥ 1

We deduce from theorem 2.12 that

\begin{matrix}h_{C_{l}}(m) = & δ_{l} m + 1 \\ h_{C_{1}∪...∪C_{l−1}}(m) = & (δ_{1} + … + δ_{l−1}) m + (l − 1) \end{matrix}

and so C_{l} ⊆ L(C_{l}) is a rational normal curve, and by the inductive hypothesis C_{1} ∪ … ∪ C_{l−1} is projectively equivalent to C(n, (deg C_{1}, …, deg C_{l−1})). Thus

h_{C}(m) = |δ|m + l − h_{I(C_{1}∪...∪C_{l−1})+I(C_{l})}(m) m ≥ 1

and so

I(C_{1} ∪ … ∪ C_{l−1}) + I(C_{l}) = (x_{0}, …, x_{n})

Then C_{1} ∪ … ∪ C_{l−1}, C_{l} lie in disjoint linear spaces by Lemma 2.17, and so C is projectively equivalent to C(n, (deg C_{1}, …, deg C_{l})). □

Now we shall derive another upper bound for the Hilbert function of an unmixed radical ideal. The following lemma is well–known, we prove it here for lack of suitable reference.

Lemma 2.19 Let A be an integral domain, K its quotient field, L a finite separable extension of K, B the integral closure of A in L. Let η ∈ B such that L = K[η], and let f ∈ K[t] be its minimal polinomial. Then

f′(η)B ⊆ A[η]

Proof. Let M ⊆ L be an A–module. Then

M′ := {x ∈ L : Tr_{K}^{L}(xM) ⊆ A}

is called the complementary module (relative to the trace) of M [25, Ch. III, §1]. It is straightfoward that if M ⊆ B then M′ ⊇ B. We have that

A[η]′ = \frac{A[η]}{f′(η)}

(see [25, Ch. III, Prop. 2, Cor.]) and so

B ⊆ A[η]′ = \frac{A[η]}{f′(η)}

□

When A is an integrally closed domain, we have that f ∈ A[t], and so in the languaje of integral dependence theory, the last assertion says that in this case f′(η) lies in the conductor of B in A[η].

Theorem 2.20 Let k be an perfect field and I ⊆ k[x_{0}, …, x_{n}] an homogeneous unmixed radical ideal of dimension d, with d ≥ 0. Then

h_{I}(m) ≤ (_{d+1}^{m+deg I+d}) − (_{d+1}^{m+d}) m ≥ 1

Proof. We shall consider first the case when P ⊆ k̄[x_{0}, …, x_{n}] is an homogeneous prime ideal.

The field k̄ is algebraically closed, and so it is both infinite and perfect. Let y_{0}, …, y_{d}, η ∈ k̄[x_{0}, …, x_{n}] be linear forms such that

k̄[y_{0}, …, y_{d}] ↪ k̄[x_{0}, …, x_{n}]/P

is an integral inclusion of graded k̄–algebras, and such that if K, L are the quotient fields of k̄[y_{0}, …, y_{d}], k̄[x_{0}, …, x_{n}]/P respectively, then K ↪ L is separable algebraic and L = K[η].

Let A := k̄[y_{0}, …, y_{d}], B := k̄[x_{0}, …, x_{n}]/P. As a consequence of Krull’s Hauptidealsatz we have that

A[η] ≅ A[t]/(F)

where F ∈ k̄[y_{0}, …, y_{d}][t] is a non zero homogeneous polynomial. We have then

dimk̄(A[η])_{m} = h_{(F)}(m) = (_{d+1}^{m+d+1}) − (_{d+1}^{m−deg F +d+1})

We have also

A[η] ↪ B ↪ \frac{A[η]}{F′(η)}

by Lemma 2.19, and thus

(_{d+1}^{m+d+1}) − (_{d+1}^{m−deg F +d+1}) ≤ h_{P} (m) ≤ (_{d+1}^{m+deg F +d}) − (_{d+1}^{m+d}) m ≥ 1

We deduce that deg F = deg P, and so

h_{P} (m) ≤ (_{d+1}^{m+deg P +d}) − (_{d+1}^{m+d}) m ≥ 1

Now we extend this bound to the case of an unmixed ideal. We have that I^{e} is and unmixed radical ideal. Let I^{e} = ∩_{P} P be the primary decomposition of I^{e}. We have

h_{I}(m) ≤ ∑_{P} {(_{d+1}^{m+deg P +d}) − (_{d+1}^{m+d})} m ≥ 1

Then we have

h_{I}(m) ≤ ∑_{P} ∑_{i=0}^{deg P −1}(_{d}^{m+d+i}) ≤ ∑_{i=0}^{deg I−1}(_{d}^{m+d+i}) = (_{d+1}^{m+deg I+d}) − (_{d+1}^{m+d})

□

Remark 2.21 This inequality is sharp for big values of m, as it is seen by comparing it with the principal term of the Hilbert polynomial of I

¿From the expression

h_{I}(m) ≤ (_{d+1}^{m+deg I+i}) − (_{d+1}^{m+d}) = ∑_{i=0}^{deg I−1}(_{d}^{m+d+i})

we see that it does not improve Chardin’s estimate [10, Th.]

h_{I}(m) ≤ deg I (_{d}^{m+d}) = ∑_{i=0}^{deg I−1}(_{d}^{m+d})

in any case. However we remark that the proof is simpler and that we can use it in our aplications instead of Chardin’s estimate obtaining very similar results.

Let k be a perfect field, I ⊆ k[x_{0}, …, x_{n}] an homogeneous unmixed radical ideal of dimension d ≥ 0, and let H_{I} be its Hilbert–Poincaré series. Then the previous result states that

t^{deg I−1}H_{I}(t) ≤ \frac{1 − t^{deg I}}{(1 − t)^{d+2}}

in the sense that this inequality holds for each term of the power series.

We derive an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical ideal, which need not be unmixed nor radical. This result is an application of both our upper and lower bounds for the Hilbert function. The use of our upper bound (Theorem 2.20) can be replaced by Chardin’s estimate [10, Th.] but the bound so obtained is essentially the same. In this way we keep our exposition self–contained.

Lemma 2.22 Let k be a perfect field and I ⊆ k[x_{0}, …, x_{n}] an homogeneous unmixed radical ideal of dimension d, with d ≥ 1, and let η ∈ k[x_{0}, …, x_{n}] be a linear from which is a non zero divisor modulo I. Then there exists m_{0} such that

h_{(I,η)}(m_{0}) ≤ (_{d}^{m_{0}+d}) − (_{d}^{m_{0}+d−3 deg I})

and 3 deg I ≤ m_{0} ≤ 5 d deg I.

Proof. Let δ := deg I, k := 3 δ, l := 2 δ, m := 5 d δ. We aim at proving that

∑_{j=0}^{l−1}{(_{d}^{m−j+d}) − (_{d}^{m−j+d−k})} ≥ ∑_{j=0}^{l−1} h_{(I,η)}(m − j)

We have that

∑_{j=0}^{l−1}{(_{d}^{m+d−j}) − (_{d}^{m+d−k−j})} = {(_{d+1}^{m+d+1}) − (_{d+1}^{m+d+1−l})} − {(_{d+1}^{m+d+1−k}) − (_{d+1}^{m+d+1−k−l})}

We have also that

∑_{j=0}^{l−1} h_{(I,η)}(m − j) = h_{(I,η^{r})}(m) ≤ {(_{d+1}^{m+d+δ}) − (_{d+1}^{m+d})} − {(_{d+1}^{m+d+1−l}) − (_{d+1}^{m+d+1−δ−l)})}

by application of Theorems 2.20 and 2.4. Thus it suffices to prove that

\begin{aligned}\{(_{d+1}^{m+d+1−δ}) − (_{d+1}^{m+d+1−δ−l})\} & −\{(_{d+1}^{m+d+1−k}) − (_{d+1}^{m+d+1−k−l})\} ≥ \\ & ≥ \{(_{d+1}^{m+d+δ}) − (_{d+1}^{m+d})\} − \{(_{d+1}^{m+d+1}) − (_{d+1}^{m+d+1−δ})\} \end{aligned}

We have that

{(_{d+1}^{m+d+1−δ}) − (_{d+1}^{m+d+1−δ−l})} −{(_{d+1}^{m+d+1−k}) − (_{d+1}^{m+d+1−k−l})} =
= ∑_{i=1}^{l}{(_{d}^{m+d+1−δ−i}) − (_{d}^{m+d+1−k−i})} =
= ∑_{i=1}^{l} ∑_{j=1}^{k−δ} (_{d−1}^{m+d+1−δ−i−j}) ≥ l (k − δ) (_{d−1}^{m+d−1−k−l})

and

\begin{aligned}\{(_{d+1}^{m+d+δ}) − (_{d+1}^{m+d})\} − \{(_{d+1}^{m+d+1}) & − & (_{d+1}^{m+d+1−δ})\} = ∑_{i=1}^{δ}\{(_{d}^{m+d+δ−i}) − (_{d}^{m+d+1−i})\} = \\ & = & ∑_{i=1}^{δ} ∑_{j=1}^{δ} (_{d−1}^{m+d+δ−i−j}) ≤ δ^{2} (_{d−1}^{m+d−1+δ}) \end{aligned}

and thus it suffices to prove that

4 = \frac{l (k − δ)}{δ^{2}} ≥ \frac{(_{d−1}^{m+d−1+δ})}{(_{d−1}^{m+d−1−k−l})}

This is clear when d = 1, as in this case the right side of this expression equals 1. When d ≥ 2 we have that

\frac{(_{d−1}^{m+d−1+δ})}{(_{d−1}^{m+d−1−k−l})} = ∏_{j=1}^{d−1} \frac{m + δ + j}{m − k − l + j} ≤ (1 + \frac{6/5}{d − 1})^{d−1} ≤ e^{\frac{6}{5}}

and so our claim follows, and we conclude that

h_{(I,η)}(m_{0}) ≤ (_{d}^{m_{0}+d}) − (_{d}^{m_{0}+d−3 deg I})

for some m_{0} such that 5 d δ − 2 δ + 1 ≤ m_{0} ≤ 5 d δ.

□

Theorem 2.23 Let k be a perfect field and I ⊆ k[x_{0}, …, x_{n}] be an homogeneous unmixed radical ideal of dimension d, with d ≥ 0, and let f ∈ k[x_{0}, …, x_{n}] be a non zero divisor modulo I. Then

h_{(I,f)}(m) ≤ deg I  m ≥ 1
h_{(I,f)}(m) = 0  m ≥ deg I + deg f − 1

if d = 0 and

h_{(I,f)}(m) ≤ 3 deg f deg I (_{d−1}^{m+d−1})

if d ≥ 1 and m ≥ 5 d deg I.

Proof. Let δ := deg I, d_{0} := deg f. We have

h_{(I,f)}(m) = h_{I}(m) − h_{I}(m − d_{0})

Consider first the case d = 0. Then h_{I}(m) ≤ δ for m ≥ 1 and h_{I}(m) = δ for m ≥ δ − 1 by Lemma 2.3, and thus

h_{(I,f)}(m) = 0 m ≥ δ + d_{0} − 1

Now let d ≥ 1. We have that I^{e} is an unmixed radical ideal, and so there exists a linear form η ∈ k[x_{0}, …, x_{n}] which is a non zero divisor modulo I^{e}. By Lemma 2.22

h_{(I^{e},η)}(m_{0}) ≤ (_{d}^{m+d}) − (_{d}^{m+d−3 deg I})

for some 3 deg I ≤ m_{0} ≤ 5 d deg I.

Let m ≥ 3 δ. We have then that

(_{d}^{m+d}) − (_{d}^{m+d−3 deg I}) = ∑_{j=1}^{3 δ} (_{m−j+1}^{m+d−j})

is the m–binomial expansion of (_{d}^{m+d}) − (_{d}^{m+d−3 deg I}), and so

h_{(I,η)}(m) ≤ (_{d}^{m+d}) − (_{d}^{m+d−3 δ})

for m ≥ m_{0} by Macaulay’s theorem and Remark 1.2. We have then

h_{(I,f)}(m) = h_{(I^{e},f)}(m) = ∑_{j=0}^{d_{0}−1} h_{(I^{e},η)}(m − j) ≤ 3 d_{0} δ (_{d−1}^{m+d−1})

for m ≥ 5 d δ.

□

§ 3. Construction of Regular Sequences

In this section we devote to the construction of regular sequences with polynomials of controlled degrees satisfying different conditions. Throughout this section k will denote an infinite perfect field

An upper bound for the Hilbert function implies in rather a direct way the existence of regular sequences in the ideal I with polynomials of bounded degree. The estimations we get are somewhat worse than [10, §2, Cor. 2 and Cor. 4]. Compare also with [28, §1, Cor. 1] and [19, Prop. 3].

Lemma 3.24 Let I, P ⊆ k[x_{0}, …, x_{n}] be homogeneous ideals, I unmixed radical of dimension d, d ≥ 0, P prime of dimension e, with e > d. Then there exists f ∈ I − P such that

deg f ≤ (e! deg I)^{\frac{1}{e−d}}

Proof. We have

h_{P} (m) ≥ (_{e}^{m+e})

Let δ := deg I, m_{0} := [(e! δ)^{\frac{1}{e−d}}]. Then

\begin{aligned}e! (_{e}^{m_{0}+e}) & ≥ (m_{0} + 1)(m_{0} + 2)^{e−1} > \\ & > (e! δ)^{\frac{e}{e−d}} + e! δd ≥ e! δ (m_{0}^{d} + d) \end{aligned}

as e ≥ 1. By Theorem 2.12 we have h_{P} (m_{0}) > h_{I}(m_{0}), and so there exists f ∈ I −P such that deg f = m_{0}. □

Proposition 3.25 Let I, J ⊆ k[x_{0}, …, x_{n}] be homogeneous ideals, I unmixed radical of dimension d, d ≥ 0, J Cohen–Macaulay of dimension e, with e ≥ d. Then there exist homogeneous polynomials f_{1}, …, f_{e−d} ∈ I such that

deg f_{j} ≤ ((d + j)! deg I)^{\frac{1}{j}}

for j = 1, …, e − d, whose image in k[x_{0}, …, x_{n}]/J form a regular sequence.

Proof. We proceed by induction on the dimension of J. If dim J = dim I = d there is nothing to prove.

Let dim J = e with d + 1 ≤ e ≤ n. Let P be an associated prime of J. The ideal J is unmixed, and so dim P = e, and thus by Lemma 3.24 there exists f_{P} ∈ I − P such that

deg f_{P} ≤ (e! deg I)^{\frac{1}{e−d}}

By eventually multiplying each f_{P} by a linear form which is a non zero divisor modulo P we can suppose that

deg f_{P} = [(e! deg I)^{\frac{1}{e−d}}]

for each associated prime ideal P of J. As the field is infinite, there exists a linear combination

f := ∑_{P ∈ Ass(J)} λ_{P} f_{P}

such that f ∈ I − P for every associated prime ideal of J. Then f is homogeneous,

deg f = [(e! deg I)^{\frac{1}{e−d}}]

and it is a non divisor of zero modulo J. Thus (J, f) is a Cohen–Macaulay ideal of dimension e − 1, and we can apply the inductive hypothesis to get homogeneous polynomials f_{1}, …, f_{e−d−1} ∈ I which form a regular sequence in R/(J, f) with deg f_{j} ≤ ((d + j)! deg I)^{\frac{1}{j}} for j = 1, …, e − d − 1. Thus f, f_{1}, …, f_{e−d−1} are homogeneous polynomials which form a regular sequence in R/J, and so

f_{1}, …, f_{e−d−1}, f ∈ R/J

is a regular sequence for which it holds the stated bounds on the degrees. □

Consider an homogeneous unmixed radical ideal I ⊆ k[x_{0}, …, x_{n}] of dimension d ≥ 0, and an homogeneous polynomial F ∈ k[x_{0}, …, x_{n}] which is a nonzero divisor modulo I. We shall show first that there exist homogeneous polynomials of controlled degrees f_{1}, …, f_{n−d} ∈ I which form a regular sequence which avoids the hypersurface {F = 0}, i. e. such that no associated prime ideal of (f_{1}, …, f_{i}) lies in {F = 0} for 1 ≤ i ≤ n−d. This result is an application of our bound for the Hilbert function of a generic hypersurface section of an unmixed radical ideal (Theorem 2.23).

Lemma 3.26 Let I, P ⊆ k[x_{0}, …, x_{n}] be homogeneous ideals, I unmixed radical of dimension d, with d ≥ 0, P prime of dimension e, with e ≥ d. Let F ∈ k[x_{0}, …, x_{n}] be an homogeneous polynomial which is a non zero divisor modulo I. Then there exists g ∈ (I, F) − P such that

deg g ≤ deg I + deg F − 1  if d=0
deg g ≤ 5 d deg F deg I  if d ≥ 1.

Proof. Let δ := deg I, d_{0} := deg F, J := (I, F). Consider first the case d = 0. Then

h_{J}(m) = 0 m ≥ δ + d_{0} − 1

by Theorem 2.23, and so there exist g ∈ J − P with deg g ≤ δ + d_{0} − 1.

Now let d ≥ 1. We have that P ⊆ k[x_{0}, …, x_{n}] is an homogeneous prime ideal of dimension e ≥ d, and so

h_{P} (m) ≥ (_{e}^{m+e}) ≥ (_{d}^{m+d})

Let m_{0} := 5 d d_{0} δ. We have then

h_{(I,f)}(m_{0}) ≤ 3 d_{0} δ (_{d−1}^{m_{0}+d−1}) < (_{d}^{m_{0}+d}) ≤ h_{P} (m)

by Theorem 2.23, and so there exists g ∈ J − P such that deg g ≤ m_{0}. □

Theorem 3.27 Let I ⊆ k[x_{0}, …, x_{n}] be an unmixed radical ideal of dimension d, with d ≥ 0, and let F ∈ k[x_{0}, …, x_{n}] be an homogeneous polynomial which is a non zero divisor modulo I. Then there exist homogeneous polynomials f_{1}, …, f_{n−d} ∈ I such that F, f_{1}, …, f_{n−d} ∈ k[x_{0}, …, x_{n}] is a regular sequence and

deg f_{i} ≤ deg I + deg F − 1  if d = 0
deg f_{i} ≤ 5 d deg F deg I  if d ≥ 1.

Proof. Applying Lemma 3.26 it follows that there exist homogeneous polynomials g_{1}, …, g_{n−d} ∈ (I, F) such that F, g_{1}, …, g_{n−d} is a regular sequence and

deg g_{i} ≤ deg I + deg F − 1  if d = 0
deg g_{i} ≤ 5 d deg F deg I  if d ≥ 1.

The proof this statement is similar to that of Proposition 3.25, we omit it here in order to avoid repetitive arguments.

Let g_{i} = f_{i} + F h_{i} with f_{i} ∈ I for 1 ≤ i ≤ n − d. Then deg f_{i} = deg g_{i}, and g_{i} ≡ f_{i} mod (F), and so F, f_{1}, …, f_{n−d} is a regular sequence for which it holds the announced bounds on the degrees. □

We observe that in the case when deg F = 1, Lemma 3.26 can be deduced from Lemma 2.22, and so both Lemma 3.26 and Theorem 3.27 do not depend on Macaulay theorem. It can also be shown in the case when deg F ≥ 2 that they do not depend on Macaulay theorem altogether [32, Th. 3.40].

Definition 3.28 Let A be a ring. Then f_{1}, …, f_{s} ∈ A is a weak regular sequence if f̄_{i} is a non zero divisor in A/(f_{1}, …, f_{i−1}) for 1 ≤ i ≤ s.

This definition differs from the definition of regular sequence in that we allow f̄_{s} ∈ A/(f_{1}, …, f_{s−1}) to be a unit.

Let F, f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}] be homogeneous polynomials such that f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}]_{F} is a weak regular sequence. Then it is not always the case that f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}] is a regular sequence, as some components of high dimension may appear in the hypersurface {F = 0}. Consider the following example:

Example 3.29 Let F ∈ k[x_{0}, x_{1}, x_{2}, x_{3}] be an homogeneous polynomial of degree d ≥ 1 such that F ∉ (x_{1}, x_{2}). Let

f_{1} := x_{1}, f_{2} := x_{1}^{d+1} + x_{2}F, f_{3} := x_{1}^{d+1} + x_{3}F ∈ k[x_{0}, x_{1}, x_{2}, x_{3}]

Then f_{2} ≡ x_{2}F, f_{3} ≡ x_{3}F mod (f_{1}), and so they form a regular sequence in k[x_{0}, x_{1}, x_{2}, x_{3}]_{F}. We have that

{(x_{0} : …: x_{4}) ∈ IP^{3} : F = 0, x_{1} = 0} ⊆ V (f_{1}, f_{2}, f_{3}) ⊆ IP^{3}

and so f_{1}, f_{2}, f_{3} ∈ k[x_{0}, x_{1}, x_{2}, x_{3}] cannot be a regular sequence.

We shall show that the weak regular sequence f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}]_{F} can in fact be replaced by polynomials p_{1}, …, p_{s} ∈ k[x_{0}, …, x_{n}] of controlled degrees such that (f_{1}, …, f_{i}) = (p_{1}, …, p_{i}) ⊆ k[x_{0}, …, x_{n}]_{F} for 1 ≤ i ≤ s and such that p_{1}, …, p_{s} ∈ k[x_{0}, …, x_{n}] is a regular sequence. Our proof follows T. Dubé’s arguments, who gave an incomplete proof of a similar statement [11, Lemma 4.1] under an unproved assumption on the Hilbert function of a certain class of ideals [11, §2.1].

Proposition 3.30 Let s ≤ n + 1, and let F, f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}] be homogeneous polynomials, with deg F ≥ 1, such that f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}]_{F} is a weak regular sequence and such that (f_{1}, …, f_{i}) ∈ k[x_{0}, …, x_{n}]_{F} is a radical ideal for 1 ≤ i ≤ s − 1. Let I_{i} := (f_{1}, …, f_{i}) ⊆ k[x_{0}, …, x_{n}]_{F} and let I_{i}^{c} := I_{i} ∩ k[x_{0}, …, x_{n}] for 1 ≤ i ≤ s. Then there exist homogeneous polynomials p_{1}, …, p_{s} ∈ k[x_{0}, …, x_{n}] which satify the following conditions:

i) p_{1} = F^{c_{1}}f_{1}, p_{2} = F^{c_{2}}f_{2}, p_{i} ≡ F^{c_{i}}f_{i} mod I_{i−1} with c_{i} ∈ ZZ, for i = 1, …, s.

ii) p_{1}, …, p_{s} ∈ k[x_{0}, …, x_{n}] is a regular sequence.

iii) deg p_{i} ≤ max {deg f_{i}, 5(n + 1 − i) deg F deg I_{i−1}^{c}} if i ≤ n

deg p_{n+1} = max {deg f_{n+1}, deg I_{n}^{c} + deg F − 1}

Proof. We shall proceed by induction. Let f_{1} = F^{e_{1}}a_{1}, f_{2} = F^{e_{2}}a_{2}, with F |/a_{1}, F |/a_{2}. Then f_{1}, f_{2} is a regular sequence in k[x_{0}, …, x_{n}]_{F} if and only if (a_{1} : a_{2}) = 1, and thus

p_{1} := F^{−e_{1}}f_{1}, p_{2} := F^{−e_{2}}f_{2}

is a regular sequence in k[x_{0}, …, x_{n}]. Now let i ≥ 3, and suppose that p_{1}, …, p_{i−1} are already constructed with the stated properties. Let

L_{i−1} := (p_{1}, …, p_{i−1}) ⊆ k[x_{0}, …, x_{n}]

and let L_{i−1} = ∩_{j=1}^{t}Q_{j} be an irredundant primary decomposition of L_{i−1} such that

\begin{matrix}F ∉ \sqrt{Q_{j}} & for 1 ≤ j ≤ r \\ F ∈ \sqrt{Q_{j}} & for r + 1 ≤ j ≤ t \end{matrix}

Let (L_{i−1}) ⊆ k[x_{0}, …, x_{n}]_{F}. Then (L_{i−1}) = I_{i−1} and so

I_{i−1}^{c} = ∩_{j=1}^{r}Q_{j} ⊆ k[x_{0}, …, x_{n}]

is a primary decomposition of I_{i−1}^{c}. We have I_{i−1}^{c} = (1) or dim I_{i−1}^{c} = n−i+1. Then there exist b_{1}, …, b_{i−1} ∈ I_{i−1}^{c} homogeneous polynomials such that F, b_{1}, …, b_{i−1} is a regular sequence and such that

deg b_{j} = max{deg f_{i}, 5 (n + 1 − i) deg F deg I_{i−1}^{c}} 1 ≤ j ≤ i − 1

if i ≤ n and

deg b_{j} = max{deg f_{n+1}, deg I_{n}^{c} + deg F − 1} 1 ≤ j ≤ n

if i = n + 1, by application of Theorem 3.27 and by eventually multiplying each b_{j} by an appropiated linear form. Let

u_{i} := ∑_{j=1}^{i−1} λ_{j} b_{j} ∈ I_{i−1}^{c}

be a linear combination of the b_{j}. We shall prove that a generic choice of λ_{1}, …, λ_{i−1} makes p_{i} := F^{c_{i}}f_{i} + u_{i} with c_{i} := deg u_{i} − deg f_{i} ≥ 0 satisfy the stated conditions. We have

deg p_{i} = max{deg f_{i}, 5 (n + 1 − i) deg F deg I_{i−1}^{c}}, if i ≤ n and

deg p_{n+1} = max{deg f_{n+1}, deg I_{n}^{c} + deg F − 1}

We aim at proving that p_{i} does not belong to any of the associated prime ideals of L_{i−1}.

Consider first 1 ≤ j ≤ r. Then f_{i} ∉ \sqrt{Q_{j}} as f_{i} is a non zero divisor modulo I_{i−1}.

We have that u_{i} ∈ I_{i−1}, and so p_{i} = F^{c_{i}}f_{i} + u_{i} ∉ \sqrt{Q_{j}}.

Now let r + 1 ≤ j ≤ t. Then dim Q_{j} = n − i + 1 as L_{i−1} is an unmixed ideal of dimension n − i + 1, and we have also F ∈ \sqrt{Q}_{j}. Thus there exist 1 ≤ l ≤ i − 1 such that b_{l} ∉ \sqrt{Q}_{j}, and so p_{i} ∉ \sqrt{Q}_{j} for a generic choice of the λ_{1}, …, λ_{i−1}. □

As a corollary, we deduce that if we have a weak regular sequence f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] of afine polynomials, we can replace it by another weak regular sequence p_{1}, …, p_{s} ∈ k[x_{1}, …, x_{n}] with polynomials of controlled degrees such that (f_{1}, …, f_{i}) = (p_{1}, …, p_{i}) for 1 ≤ i ≤ s and such that the homogenizated polynomials p̃_{1}, …, p̃_{s} ∈ k[x_{0}, …, x_{n}] form a regular sequence.

Corollary 3.31 Let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be a weak regular sequence of affine polynomials such that (f_{1}, …, f_{i}) ⊆ k[x_{1}, …, x_{n}] is a radical ideal for 1 ≤ i ≤ s − 1. Let I_{i} := (f_{1}, …, f_{i}) ⊆ k[x_{1}, …, x_{n}] for 1 ≤ i ≤ s. Then there exist p_{1}, …, p_{s} ∈ k[x_{1}, …, x_{n}] which satisfy the following conditions: i) p_{1} = f_{1}, p_{2} = f_{2}, p_{i} ≡ f_{i} mod I_{i−1} for i = 1, …, s.

ii) p̃_{1}, …, p̃_{s} ∈ k[x_{0}, …, x_{n}] is a regular sequence.

iii) deg p_{i} ≤ max {deg f_{i}, 5(n + 1 − i) deg Ĩ_{i−1}} if i ≤ n

deg p_{n+1} = max{deg f_{n+1}, deg Ĩ_{n}}

Proof. We have that f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] is a weak regular sequence, and so f̃_{1}, …, f̃_{s} ∈ k[x_{0}, …, x_{n}]_{x_{0}} is also a weak regular sequence, and also (f̃_{1}, …, f̃_{i}) ⊆ k[x_{0}, …, x_{n}]_{x_{0}} is radical for 1 ≤ i ≤ s − 1. Let r_{1}, …, r_{s} ∈ k[x_{0}, …, x_{n}] be the homogeneous polynomials obtained by aplying Proposition 3.30 to f̃_{1}, …, f̃_{s}. Let

p_{i} := r_{i}^{a} 1 ≤ i ≤ s

Thus deg p_{i} ≤ deg r_{i}, and x_{0}^{e_{i}} p̃_{i} = r_{i} for some e_{i} ≥ 0. Then p̃_{1}, …, p̃_{s} ∈ k[x_{0}, …, x_{n}] is a regular sequence, and p_{1}, …, p_{s} satisfy the stated conditions. □

Our bounds for the degrees in the preceding propositions depend on the degree of certain ideals associated to f_{1}, …, f_{s}. The following is a Bézout–type lemma which shows that these bounds can also be expressed in terms of the degrees of the polynomials f_{1}, …, f_{s}.

Lemma 3.32 Let s ≤ n, and let F, f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}] be homogeneous polynomials, with deg F ≥ 1, such that f_{1}, …, f_{s} ∈ k[x_{0}, …, x_{n}]_{F} is a weak regular sequence. Let I := (f_{1}, …, f_{s}) ⊆ k[x_{0}, …, x_{n}]_{F}, and let I^{c} := I ∩ k[x_{0}, …, x_{n}]. Then

deg I^{c} ≤ ∏_{i=1}^{s} deg f_{i}

Proof. If I^{c} = (1) there is nothing to prove. Otherwise we have dim I^{c} ≥ 0.

Let I_{i} := (f_{1}, …, f_{i}) ⊆ k[x_{0}, …, x_{n}]_{F}, J_{i} := (I_{i−1}^{c}, f_{i}) ⊆ k[x_{0}, …, x_{n}] for 1 ≤ i ≤ s. Then dim I_{i}^{c} = dim J_{i} = n − i and J_{i} ⊆ I_{i}^{c}, and so deg I_{i}^{c} ≤ deg J_{i}.

We shall proceed by induction on i. For i = 1 we have

deg I_{1}^{c} ≤ deg J_{1} = deg f_{1}

Let i ≥ 2. Then f_{i} is a non zero divisor modulo I_{i−1}^{c} and so

deg I_{i}^{c} ≤ deg J_{i} = deg f_{i} deg I_{i−1}^{c} ≤ ∏_{j=1}^{s} deg f_{j}

by the inductive hypothesis. □

§ 4. The Effective Nullstellensatz and the Representation Problem in Complete Intersections

In this section we consider the problem of bounding the degrees of the polynomials in the Nullstellensatz and in the representation problem in complete intersections.

As a consequence of the results of the previous section we obtain bounds for these two problems which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the geometric degree of the system of equations. The bounds so obtained are more intrinsic and refined than the usual estimates, and we show that they are sharper in some special cases.

Our arguments at this point are essentially the same of T. Dubé [11].

The bound we obtain for the effective Nulltellensatz is similar to that announced in [15, Th. 19] and proved in [14] by algorithmic methods and to that obtained in [24] by duality methods.

Let g, f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be polynomials such that g ∈ (f_{1}, …, f_{s}). Let D ≥ 0. Then there exist a_{1}, …, a_{s} ∈ k[x_{1}, …, x_{n}] such that

g = a_{1}f_{1} + … + a_{s}f_{s}

with deg a_{i}f_{i} ≤ deg g + D for i = 1, …, s if and only if

x_{0}^{D}g̃ ∈ (f̃_{1}, …, f̃_{s}) ⊆ k[x_{0}, …, x_{n}]

and so in this situation we aim at bounding D such that x_{0}^{D}g̃ ∈ (f̃_{1}, …, f̃_{s}).

We shall suppose n, s ≥ 2, as the cases n = 1 or s = 1 are well–known. Also we shall suppose without loose of generality that k is algebraically closed, and in particular infinite and perfect.

Let h_{1}, …, h_{s} ∈ k[x_{1}, …, x_{n}] be a weak regular sequence of affine polynomials such that (h_{1}, …, h_{i}) is radical for 1 ≤ i ≤ s − 1. In particular we have s ≤ n + 1. We fix the following notation:

\begin{aligned}I_{i} & := & (h_{1}, …, h_{i}) ⊆ k[x_{1}, …, x_{n}] \\ J_{i} & := & (Ĩ_{i−1}, h̃_{i}) ⊆ k[x_{0}, …, x_{n}] \\ H_{i} & := & (h̃_{1}, …, h̃_{s}) ⊆ k[x_{0}, …, x_{n}] \end{aligned}

for 1 ≤ i ≤ s. Let J_{i} = ∩_{P} Q_{P} be a primary decomposition of J_{i}, and let

J_{i}^{∗} = ⋂_{P:dim P =dim I} Q_{P}

be the intersection of the primary components of maximal dimension of J_{i}, which is well defined as the isolated components are unique. We have that J_{i} ⊆ J_{i}^{∗} ⊆ Ĩ_{i}.

Let

γ_{1} := 0
γ_{i} := deg h_{i} deg Ĩ_{i−1} − deg Ĩ_{i}  1 ≤ i ≤ n,
γ_{n+1} := deg h_{n+1} + deg Ĩ_{n} − 1

Proposition 4.33 Let g ∈ Ĩ_{i} for some 1 ≤ i ≤ s. Then

x_{0}^{γ_{i}}g ∈ J_{i}^{∗}

Proof. The case 1 ≤ i ≤ n is [11, Lemma 5.5].

We consider the case i = n + 1. We have that Ĩ_{n} ⊆ k[x_{0}, …, x_{n}] is an unmixed radical ideal of dimension 0 and we have that h_{n+1} is a non zero divisor modulo I_{n}, and so by Theorem 2.23 h_{J_{n+1}}(m) = 0 for m ≥ deg Ĩ_{n} + deg h_{n+1} − 1. Then x_{0}^{γ_{n+1}} ∈ J_{n+1} ⊆ J_{n+1}^{∗} and thus

x_{0}^{γ_{n+1}}g ∈ J_{n+1}^{∗}
□

Then we apply Corollary 3.31 to the sequence h_{1}, …, h_{s}, to obtain p_{1}, …, p_{s} ∈ k[x_{1}, …, x_{n}] such that

i) p_{1} = h_{1}, p_{2} = h_{2}, p_{i} = h_{i} + u_{i}, with u_{i} ∈ I_{i−1} for 1 ≤ i ≤ s.

ii) p̃_{1}, …, p̃_{s} ∈ k[x_{0}, …, x_{n}] is a regular sequence.

iii) deg p_{i} ≤ max {deg h_{i}, 5 (n + 1 − i) deg Ĩ_{i−1}} for 1 ≤ i ≤ n,

deg p_{n+1} ≤ max {deg h_{n+1}, deg Ĩ_{n}}

Then p̃_{i} = x_{0}^{c_{i}} f̃_{i}+ũ_{i}, with c_{1} = 0, c_{2} = 0, c_{i} = max {0, 5 (n+1−i) deg Ĩ_{i−1}−deg h_{i}} for 1 ≤ i ≤ n, and c_{n+1} = max {0, deg Ĩ_{n} − deg h_{n+1}}. Let

D_{i} := ∑_{j=2}^{i}(i + 1 − j)γ_{j} + ∑_{j=3}^{i−1}(i − j)c_{j}

for 1 ≤ i ≤ s.

Proposition 4.34 Let g ∈ Ĩ_{i} for some 1 ≤ i ≤ s. Then

x_{0}^{D_{i}}g ∈ H_{i}

Proof. This proposition follows from the proof of [11, Lemma 6.1] and [11, Lemma 6.2], applying Proposition 4.33 for the case i = n + 1. □

Now the task consists in bounding D_{s}. Our bound will depend not only on the number of variables and on the degrees of the polynomials h_{1}, …, h_{s}, but also on the degree of some homogeneous ideals associated to them.

Proposition 4.35 Let d := max_{1≤i≤s} deg h_{i} and δ_{i} := deg Ĩ_{i} for 1 ≤ i ≤ s. We have then

i) D_{s} ≤ s^{2} (d − 1 + 3n) max_{1≤i≤s−1} δ_{i} for s ≤ n and

ii) D_{n+1} ≤ n^{2}(d − 1 + 3n) max_{1≤i≤s−1} δ_{i} Proof. Let d_{i} := deg h_{i} for 1 ≤ i ≤ s. We have

∑_{j=3}^{s−1}(s − j)c_{j} = ∑_{j=3}^{s−1}(s − j) max{0, 5(n + 1 − j)δ_{j−1} − d_{j}} ≤
≤ 5 (∑_{j=3}^{s−1}(s − j)(n + 1 − j)) max_{1≤i≤s−2} δ_{i} ≤
≤ 5 (n − 2)(∑_{j=3}^{s−1}(s − j)) max_{1≤i≤s−2} δ_{i} ≤
≤ 3(n − 2)(s − 2)^{2} max_{1≤i≤s−2} δ_{i}

Let s ≤ n. We have then

\begin{aligned}∑_{j=2}^{s}(s − j + 1) γ_{j} & = & ∑_{j=2}^{s}(s − j + 1) (d_{j}δ_{j−1} − δ_{j}) ≤ \\ & ≤ & (∑_{j=1}^{s−1} j) d max_{1≤i≤s−1} δ_{i} ≤ s^{2} d max_{1≤i≤s−1} δ_{i} \end{aligned}

Thus

D_{s} ≤ s^{2} d max_{1≤i≤s−1} δ_{i} + 3(n − 2)(s − 2)^{2} max_{1≤i≤s−2} δ_{i} ≤
≤ s^{2} (d − 1 + 3n) max_{1≤i≤s−1} δ_{i}

Also we have

\begin{aligned}∑_{j=2}^{n+1}(n + 2 − j) γ_{j} & = & ∑_{j=2}^{n}(n + 2 − j)(d_{j}δ_{j−1} − δ_{j}) + d_{n+1} + δ_{n} − 1 ≤ \\ & ≤ & (∑_{j=1}^{n} j) d max_{1≤i≤n−1} δ_{i} ≤ n^{2} d max_{1≤i≤n−1} δ_{i} \end{aligned}

and thus

D_{n+1} ≤ n^{2} d max_{1≤i≤n−1} δ_{i} + 3(n − 2)(n − 1)^{2} max_{1≤i≤n−1} δ_{i} ≤
≤ n^{2}(d − 1 + 3n) max_{1≤i≤n−1} δ_{i}
□

Given f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] polynomials which define a proper ideal (f_{1}, …, f_{s}) ⊆ k[x_{1}, …, x_{n}] of dimension n − s or 1 ∈ (f_{1}, …, f_{s}), there exist t ≤ s and h_{1}, …, h_{t} ∈ k[x_{1}, …, x_{n}] linear combinations of the polynomials {f_{i}, x_{j}f_{i} : 1 ≤ i ≤ s, 1 ≤ j ≤ n} such that

i) (h_{1}, …, h_{t}) = (f_{1}, …, f_{s})

ii) h_{1}, …, h_{t} is a weak regular sequence. iii) (h_{1}, …, h_{i}) ⊆ k[x_{1}, …, x_{n}] is a radical ideal for 1 ≤ i ≤ t − 1.

In the case when k is a zero characteristic field, we can take h_{1}, …, h_{t} linear combinations of f_{1}, …, f_{s}. In fact, in both cases a generic linear combination will satisfy the stated conditions. This result is a consequence of Bertini’s theorem [21, Cor. 6.7] (see for instance [30, §5.2], [23, Prop. 37]), and allows us to reduce from the general situation to the previously considered one.

Let d := max_{1≤i≤s} deg f_{i}, and suppose that deg f_{i} ≤ deg f_{i+1} for 1 ≤ i ≤ s − 1. Thus in the case when k is a zero characteristic field we can take h_{1}, …, h_{t} such that

deg h_{i} ≤ deg f_{i} 1 ≤ i ≤ t

and deg h_{i} ≤ d + 1 in the case when char (k) = p > 0.

Definition 4.36 Let k be a zero characteristic field and f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] polynomials which define a proper ideal (f_{1}, …, f_{s}) ⊆ k[x_{1}, …, x_{n}] of dimension n − s or such that 1 ∈ (f_{1}, …, f_{s}). For λ = (λ_{ij})_{ij} ∈ k̄^{s×s} and 1 ≤ i ≤ s let

g_{i}(λ) := ∑_{j} λ_{ij}f_{j} ∈ k̄[x_{1}, …, x_{n}]

be linear combinations of f_{1}, …, f_{s}. Consider the set of matrices Γ ⊆ k̄^{s×s} such that for λ ∈ Γ there exist t = t(λ) ≤ s such that (g_{1}, …, g_{t}) = (f_{1}, …, f_{s}), g_{1}, …, g_{t} is a weak regular sequence and (g_{1}, …, g_{i}) ⊆ k̄[x_{0}, …, x_{n}] is a radical ideal for 1 ≤ i ≤ t−1. Then Γ ≠ ∅, and in fact Γ contains a non empty open set U ⊆ k̄^{s×s}. Let V_{i}(λ) := V (g_{1}, …, g_{i}) ⊆ IA^{n} be the affine variety defined by g_{1}, …, g_{i} for 1 ≤ i ≤ s, and define

δ(λ) = max_{1≤i≤min{t(λ),n}−1} deg V_{i}(λ)

Then the geometric degree of the system of equations f_{1}, …, f_{s} is defined as

δ = δ(f_{1}, …, f_{s}) := min_{λ∈Γ} δ(λ)

In the case when char(k) = p > 0 we define the degree of the system of equations f_{1}, …, f_{s} in an analogous way by considering linear combinations of the polynomials f_{1}, …, f_{s}, x_{1}f_{1}, …, x_{n}f_{s}.

This definition extends [24, Def. 1] to the case of a complete intersection ideal. It is analogous to the definition of degree of a system of equations of [15], though this degree is not defined as a minimum through all the possible choices of λ ∈ Γ but through a generic choice.

Remark 4.37 We see from the definition that the degree of a system of equations f_{1}, …, f_{s} does not depend on inversible linear combinations, i. e. if μ = (μ_{ij})_{ij} ∈ GL_{s}(k) and

g_{i}(μ) := ∑_{j} μ_{ij}f_{j}

for 1 ≤ i ≤ s, then δ(f_{1}, …, f_{s}) = δ(g_{1}, …, g_{s}), and so this parameter is in some sense an invariant of the system.

The following lemma shows that δ(f_{1}, …, f_{s}) can be bounded in terms of the degrees of the polynomials f_{1}, …, f_{s}.

Lemma 4.38 Let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be polynomials which define a proper ideal (f_{1}, …, f_{s}) ⊆ k[x_{1}, …, x_{n}] of dimension n − s, or 1 ∈ (f_{1}, …, f_{s}) and such that d_{i} ≥ d_{i+1} for 1 ≤ i ≤ s − 1, with d_{i} := deg f_{i}, and let d := max_{1≤i≤s} d_{i}. Then

δ(f_{1}, …, f_{s}) ≤ ∏_{i=1}^{min{s,n}−1} d_{i}

in the case when k is a zero characteristic field, and

δ(f_{1}, …, f_{s}) ≤ (d + 1)^{min{s,n}−1}

in the case when char(k) = p > 0.

Proof. This follows at once from Lemma 3.32. □

We have the following bounds for the representation problem in complete intersections and for the effective Nullstellensatz in terms of this parameter.

Theorem 4.39 (Representation Problem in Complete Intersections) Let s ≤ n, and let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be polynomials which define a proper ideal (f_{1}, …, f_{s}) ⊆ k[x_{1}, …, x_{n}] of dimension n−s. Let d := max_{1≤i≤s} deg f_{i}, and let δ be the geometric degree of the system of equations f_{1}, …, f_{s}. Let g ∈ (f_{1}, …, f_{s}). Then there exist a_{1}, …, a_{s} ∈ k[x_{1}, …, x_{n}] such that

g = a_{1}f_{1} + … + a_{s}f_{s}

with deg a_{i}f_{i} ≤ deg g + s^{2} (d + 3n) δ for i = 1, …, s.

Proof. This follows from Proposition 4.35 (i). □

Theorem 4.40 (Effective Nullstellensatz) Let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] be polynomials such that 1 ∈ (f_{1}, …, f_{s}). Let d := max_{1≤i≤s} deg f_{i}, and let δ be the geometric degree of the system of equations f_{1}, …, f_{s}. Then there exist a_{1}, …, a_{s} ∈ k[x_{1}, …, x_{n}] such that

1 = a_{1}f_{1} + … + a_{s}f_{s}

with deg a_{i}f_{i} ≤ min{n, s}^{2} (d + 3n) δ for i = 1, …, s.

Proof. This follows from Proposition 4.35. □

We can essentially recover from Theorem 4.39 and Theorem 4.40 the usual bounds for the representation problem in complete intersections and the effective Nullstellensatz. We have for instance:

Corollary 4.41 Let k be a zero characteristic field and f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] polynomials such that 1 ∈ (f_{1}, …, f_{s}), and such that d_{i} ≥ d_{i+1} for 1 ≤ i ≤ s − 1, with d_{i} := deg f_{i}. Then there exist a_{1}, …, a_{s} ∈ k[x_{1}, …, x_{n}] such that

1 = a_{1}f_{1} + … + a_{s}f_{s}

with deg a_{i}f_{i} ≤ min{n, s}^{2} (d + 3n) ∏_{j=1}^{min{n,s}−1} d_{j} for i = 1, …, s.

□

We remark that our results yield much sharper bounds for these two problems in some particular cases. Consider for instance the following example:

Example 4.42 Let k be a zero characteristic field and h_{1}, …, h_{s} ∈ k[x_{1}, …, x_{n}] a weak regular sequence of polynomials such that 1 ∈ (h_{1}, …, h_{s}). Let d := max_{1≤i≤s} deg h_{i}, and let f_{1}, …, f_{s} ∈ k[x_{1}, …, x_{n}] such that

f_{i} = h_{i} + u_{i}

with u_{i} ∈ (h_{1}, …, h_{i−1}) for 1 ≤ i ≤ s. Then

δ := δ(f_{1}, …, f_{s}) = δ(h_{1}, …, h_{s}) ≤ d^{min{n,s}−1}

Let D := max_{1≤i≤s} deg f_{i}. By Theorem 4.40 there exist a_{1}, …, a_{s} ∈ k[x_{1}, …, x_{n}] such that

1 = a_{1}f_{1} + …, a_{s}f_{s}

with deg a_{i}f_{i} ≤ min{n, s}^{2} (D + 3n) d^{min{n,s}−1} for i = 1, …, s. This estimate is sharper for big values of D than the bound

deg a_{i}f_{i} ≤ D^{min{n,s}} i = 1, …, s

for D ≥ 3, which results from application of the bound of [22].

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