Hilbert-Samuel Function Research Articles

A note on Ratliff-Rush filtration, reduction number and postulation number of [formula omitted]-primary ideals

Let (R,m) be a Cohen-Macaulay local ring of dimension d≥2, and I an m-primary ideal. Let r(I) be the reduction number of I, n(I) the postulation number and ρ(I) the stability index of the Ratliff-Rush filtration with respect to I. We prove that for d=2, if n(I)=ρ(I)−1, then r(I)≤n(I)+2, and if n(I)≠ρ(I)−1, then r(I)≥n(I)+2. For d≥3, assuming I is integrally closed, depthgr(I)=d−2, and n(I)=−(d−3), we prove that r(I)≥n(I)+d. Our main result generalizes a result by Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring to the good behavior of the Ratliff-Rush filtration with respect to I mod a superficial sequence. From this result, it follows that for Cohen-Macaulay local rings of dimension d≥2, if PI(k)=HI(k) for some k≥ρ(I), then PI(n)=HI(n) for all n≥k.

CONSTANCY OF THE HILBERT–SAMUEL FUNCTION

Abstract We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.

Finite determinacy and approximation of flat maps

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real (resp. complex) analytic spaces to R m \mathbb {R}^m (resp. C m \mathbb {C}^m ). It is shown that the flatness of analytic maps from germs of real (resp. complex) analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection (resp. Cohen-Macaulay) analytic germs can be approximated arbitrarily well by polynomial (resp. Nash) maps in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney’s tangent cone.

Hilbert-Kunz function and Hilbert-Kunz multiplicity of some ideals of the Rees algebra

We prove that the Hilbert-Kunz function of the ideal (I, It) of the Rees algebra where I is an -primary ideal of a 1-dimensional local ring is a quasi-polynomial in e, for large e. For we calculate the Hilbert-Samuel function of the R-module and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal when I is a parameter ideal in a Cohen-Macaulay local ring of dimension proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case.

Equisingular algebraic approximation of real and complex analytic germs

We show that a Cohen-Macaulay analytic singularity can be arbitrarily closely approximated by germs of Nash sets which are also Cohen-Macaulay and share the same Hilbert-Samuel function. We also prove that every analytic singularity is topologically equivalent to a Nash singularity with the same Hilbert-Samuel function.

Estimates for the coefficients of differential dimension polynomials

We answer the following longstanding question of Kolchin: given a system of algebraic-differential equations $\Sigma (x_1,\dots ,x_n)=0$ in $m$ derivatives over a differential field of characteristic zero, is there a computable bound that only depends on the order of the system (and on the fixed data $m$ and $n$) for the typical differential dimension of any prime component of $\Sigma$? We give a positive answer in a strong form; that is, we compute a (lower and upper) bound for all the coefficients of the Kolchin polynomial of every such prime component. We then show that, if we look at those components of a specified differential type, we can compute a significantly better bound for the typical differential dimension. This latter improvement comes from new combinatorial results on characteristic sets, in combination with the classical theorems of Macaulay and Gotzmann on the growth of Hilbert-Samuel functions.

On the asymptotic behaviour of Hilbert functions of graded rings and modules: applications to the Hilbert–Samuel functions of well filtered modules, multiplicity and analytic spread of nice filtrations

This paper is devoted to the study of the asymptotic behaviour of Hilbert functions on finitely generated graded modules and particularly to some investigations on the asymptotic behaviour of Hilbert–Samuel functions on finitely generated well filtered modules. Consider a graded noetherian ring $$A=\bigoplus _{n\in \mathbb {N}}A_{n}$$ which may be assumed to be of the form $$A=A_{0}[x_{1},\ldots ,x_{s}],$$ where each element $$x_{i}$$ is homogeneous and deg $$x_{i}=$$$$k_{i}\ge 1$$ for all i. Suppose that the ring $$A_{0}$$ is artinian. Let M$$ =\bigoplus _{n\in \mathbb {Z}}M_{n}$$ be a finitely generated positively $$\mathbb {Z}$$-graded A-module. Then the length$$\ l_{A_{0}}(M_{n})$$ of the $$ A_{0}$$-module $$M_{n}$$ is finite for all $$n\in \mathbb {Z}$$. The Hilbert function $$H(M,-)$$ of M is defined as$$\ H(M,n)=$$$$\ l_{A_{0}}(M_{n})$$ for all $$n\in \mathbb {Z}$$. By a well known Hilbert Theorem, if $$k_{i}=1$$ for all i, then there exists a unique polynomial $$P(X)\in \mathbb {Q}[X]$$ with $$ \deg $$$$P(X)=d-1$$ such that $$H(M,n)=$$P(n) for all large integers n, where d denotes the Krull dimension of the A-module M. But if at least one of the homogeneous generators $$x_{i}$$ has degree $$k_{i}$$$$\ge 2,$$ then such a polynomial need not exist. Dichi and Sangare (J Pure Appl Algebra 138:205–213, 1999) had shown in that case that the Hilbert function $$H(M,-)$$ of M has a good asymptotic behaviour. More precisely, they had shown that $$H(M,-)$$ is a quasi-polynomial function. Then this result is applied, with some minor but necessary adaptation, to the graded $$G_{f}(A)$$-module $$G_{\Phi }(M)=\bigoplus _{n\text { }\ge \text { }0}\frac{M_{n}}{M_{n+1}}$$, where $$f=(I_{n})_{n\text { }\in \text { }\mathbb {Z}}$$ is a nice filtration of the ring A and where M is a finitely generated A-module which is no longer graded but which is endowed with a filtration $$\Phi =(M_{n})_{n\text { }\in \text { }\mathbb {Z}}$$ compatible with the filtration f. Here $$ G_{f}(A)=\bigoplus _{n\text { }\ge \text { }0}\frac{I_{n}}{I_{n+1}}$$ is the graded ring associated with f. This leads in particular to the definition of a concept of multiplicity $$e_{f}(M)$$ of M with respect to the filtration f which is a rational number. If the filtration f is non I-adic, then its multiplicity $$e_{f}(M) $$ need not be an integer unlikely for the multiplicity of an ideal. So it does not have any geometrical interpretation. However, it is shown in Dichi and Sangare (1999) that, for a given noetherian filtration $$f=(I_{n})_{n\text { }\in \text { }\mathbb {Z}}$$ on the noetherian finite dimensional ring A, if $$l_{A}(\frac{M}{I_{1}M})<+\infty ,$$ then the multiplicity function $$M\longmapsto e_{f}$$ (M) behaves well on short exact sequences of finitely generated A-modules. As a second application of the Hilbert Theorem, we introduce a satisfactory concept of analytic spread for filtrations and we show that this analytic spread is of “asymptotic nature” for nice filtrations.

Effective bounds for the consistency of differential equations

One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the “algebraic data” associated to the system. In technical terms, this translates to the problem of determining the existence of regular realizations of differential kernels via their possible prolongations. In this paper we effectively compute an improved upper bound for the number of prolongations needed to guarantee the existence of such realizations, which ultimately produces solutions to many types of systems of partial differential equations. This bound has several applications, including an improved upper bound for the order of characteristic sets of prime differential ideals. We obtain our upper bound by proving a new result on the growth of the Hilbert–Samuel function, which may be of independent interest.

Faîte du cône tangent à une singularité : un théorème oublié

Let X be an excellent scheme; we denote by HX the modified Hilbert–Samuel function. This function is upper semi-continuous along X and does not increase for the lexicographical ordering after permissible blowing ups. When X is embedded in a regular ambient scheme, for all x∈X, the “stable τ at x” (“τ stable de x”), denoted by τst(x), is the codimension of the ridge of the tangent cone of X at x in the tangent cone of W at x. It is well known that the functionι:X→NN×−N,x↦(HX(x),−τst(x)), does not increase for the lexicographical ordering after permissible blowing ups. In this note, we show that ι is upper semi-continuous along X. This result is generalized to the non-embedded case.

Generalized Hilbert coefficients and Northcott's inequality

Let R be a Cohen–Macaulay local ring of dimension d with infinite residue field. Let I be an R-ideal that has analytic spread ℓ(I)=d, satisfies the Gd condition and the weak Artin–Nagata property ANd−2−. We provide a formula relating the length λ(In+1/JIn) to the difference PI(n)−HI(n), where J is a general minimal reduction of I, PI(n) and HI(n) are respectively the generalized Hilbert–Samuel polynomial and the generalized Hilbert–Samuel function. We then use it to establish formulas to compute the generalized Hilbert coefficients of I. As an application, we extend Northcott's inequality to non-m-primary ideals. Furthermore, when equality holds, we prove that the ideal I enjoys nice properties. Indeed, if this is the case, then the reduction number of I is at most one and the associated graded ring of I is Cohen–Macaulay. We also recover results of G. Colomé-Nin, C. Polini, B. Ulrich and Y. Xie on the positivity of the generalized first Hilbert coefficient j1(I). Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m-primary.

Hilbert–Samuel functions of well bifiltered modules

In an earlier paper, we studied the Hilbert quasi-polynomial functions of finitely generated bigraded modules in the general framework when the base ring is bigraded and generated by finitely many homogeneous elements of arbitrary degrees. In this paper, we introduce the concept of [Formula: see text]-good bifiltration [Formula: see text] on a finitely generated [Formula: see text]-module [Formula: see text], where [Formula: see text] and [Formula: see text] are specified noetherian filtrations on the noetherian ring [Formula: see text]. The bigraded modules associated with such bifiltrations are shown to be finitely generated under reasonable hypotheses. Their Hilbert functions are studied. The Hilbert–Samuel function of [Formula: see text] with respect to the [Formula: see text]-good bifiltration [Formula: see text] of [Formula: see text] is one of them. It is proved, among others, that this function is a quasi-polynomial function in two variables and that if [Formula: see text] is a noetherian local ring and if the filtrations [Formula: see text] and [Formula: see text] are primary filtrations, then its degree equals the Krull dimension of [Formula: see text].

Irrational behavior of algebraic discrete valuations

We study algebraic discrete valuations dominating normal local domains of dimension two. We construct a family of examples to show that the Hilbert–Samuel function of the associated graded ring of the valuation can fail to be asymptotically of the form: quasi-polynomial plus a bounded function. We also show that the associated multiplicity can be irrational, or even transcendental.

The Dual Hilbert–Samuel Function of a Maximal Cohen-Macaulay Module

Let R be a Cohen–Macaulay local ring with a canonical module ω R . Let I be an 𝔪-primary ideal of R and M, a maximal Cohen–Macaulay R-module. We call the function n⟼ℓ(Hom R (M, ω R /I n+1ω R )) the dual Hilbert–Samuel function of M with respect to I . By a result of Theodorescu, this function is of polynomial type. We study its first two normalized coefficients. In particular, we analyze the case when R is Gorenstein.

Equivalence and resolution of singularities

This article provides a simple presentation of an algorithm to resolve singularities of algebraic varieties over fields of characteristic zero by means of a sequence of blowing ups with smooth centers contained in the set of points of maximum multiplicity. The algorithm uses primarily multiplicity, rather than the Hilbert–Samuel function, to control the resolution process, and it does not involve a local embedding into a smooth variety. The paper introduces a generalization of the usual notion of equivalence in the theory of resolution of singularities, which is important to justify an essential step in the construction of the algorithm.

Explicit uniform estimation of rational points I. Estimation of heights

By using the slope method in Arakelov geometry, we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic Hilbert–Samuel function.

Parallelization of Modular Algorithms

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gröbner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional ideal and describe their parallel implementation in Singular. Our modular algorithms for solving problems over Q mainly consist of three parts: solving the problem modulo p for several primes p , lifting the result to Q by applying the Chinese remainder algorithm (resp. rational reconstruction), and verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm for computing Gröbner bases can be simplified for homogeneous ideals (cf. Arnold, 2003). The idea of the proof could easily be adapted to the local case, i.e. for local orderings and not necessarily homogeneous ideals, using the Hilbert–Samuel function (cf. Pfister, 2007). In this paper we prove the corresponding theorem for non-homogeneous ideals in the case of a global ordering.

A modified coefficient ideal for use with the strict transform

Two main algorithmic approaches are known for making Hironaka’s proof of resolution of singularities in characteristic zero constructive. Their main differences are the use of different notions of transforms during the resolution process and the different use of exceptional divisors in the descent in ambient dimension. In this article, we focus on the first difference. Only the approach using the weak transform has up until now been successfully used in implementations, because the other one requires an explicit stratification by the Hilbert–Samuel function at each step of the algorithm which is highly impractical due to the high complexity of the computation of such a stratification. In this article, a (hybrid-type) algorithmic approach is proposed which allows the use of the strict transform without the full impact of the complexity of the stratification by the Hilbert–Samuel function at each step of the desingularization process. This new approach is not intended to always be superior to the previously implemented one, instead it has its strengths precisely at the weak point of the other one and is thus a candidate to be joined with it by an appropriate heuristic.

Castelnuovo–Mumford regularity and Degree of nilpotency

AbstractIn this paper we show that the Castelnuovo–Mumford regularity of the associated graded module with respect to anm-primary idealIis effectively bounded by the degree of nilpotency ofI. From this it follows that there are only a finite number of Hilbert-Samuel functions for ideals with fixed degree of nilpotency.

Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faisceaux cohérents

The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a metrized vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that were needed in previously known analogous results. Then we show how to associate an arithmetic Hilbert-Samuel function to a coherent sheaf on an arithmetic variety—provided this coherent sheaf is a subquotient of a metrized vector bundle—and, using the classical arithmetic Hilbert-Samuel theorem and our extension theorem, we give the leading term of the so constructed arithmetic Hilbert-Samuel function.

A Simplified Proof of Desingularization and Applications

This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme X is achieved by using the procedure of principalization for the ideal {\mathcal I}(X) associated to the embedded scheme X . The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.