Weierstrass Division Research Articles

Pluriharmonic Definable Functions in Some O-Minimal Expansions of the Real Field

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

Some properties of the polynomially bounded o-minimal expansions of the real field and of some quasianalytic local rings

In this paper, we study the Weierstrass division theorem over the rings of smooth germs that are definable in an arbitrary polynomially bounded o-minimal expansion of the real field by giving some criteria for satisfying this theorem. Afterwards, we study some topological properties of some quasianalytic subrings of the ring of smooth germs for the $(x_1)$-adic topology by showing that these rings are separable metric spaces. Also, we cite a criterion for their completeness with respect to the $(x_1)$-adic topology.

Effective Power Series Computations

Let K be an effective field of characteristic zero. An effective tribe is a subset of $$K [[z_1, z_2, \ldots ]] = K \cup K {[}[z_1]] \cup K [[z_1, z_2]] \cup \cdots $$ that is effectively stable under the K-algebra operations, restricted division, composition, the implicit function theorem, as well as restricted monomial transformations with arbitrary rational exponents. Given an effective tribe with an effective zero test, we will prove that an effective version of the Weierstrass division theorem holds inside the tribe and that this can be used for the computation of standard bases.

The Pila–Wilkie theorem for subanalytic families: a complex analytic approach

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^{r}$-smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a $C^{r}$ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height $H$ in a compact piece of a complex-analytic set of dimension $k$ in $\mathbb{C}^{m}$ are contained in $O(1)$ complex-algebraic hypersurfaces of degree $(\log H)^{k/(m-k)}$. This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.

Encoding Algebraic Power Series

Algebraic power series are formal power series which satisfy a univariate polynomial equation over the polynomial ring in n variables. This relation determines the series only up to conjugacy. Via the Artin-Mazur theorem and the implicit function theorem it is possible to describe algebraic series completely by a vector of polynomials in n+p variables. This vector will be the code of the series. In the paper, it is then shown how to manipulate algebraic series through their code. In particular, the Weierstrass division and the Grauert-Hironaka-Galligo division will be performed on the level of codes, thus providing a finite algorithm to compute the quotients and the remainder of the division.

Piecewise Weierstrass preparation and division for o-minimal holomorphic functions

Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the semialgebraic setting and for the structure of globally subanalytic sets and functions we obtain the corresponding results for definable real analytic functions. As an application we show a definable global Nullstellensatz for principal ideals.

Strictly convergent analytic structures

We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries [DD] on p -adic subanalytic sets, and we continue the study of non-archimedean fields with analytic structure of [LR3], [CLR1] and [CL]. We show that the language {\mathcal L}_K consisting of the language of valued fields together with all strictly convergent power series over a complete, rank one valued field K can be expanded, in a definitional way, to a larger language corresponding to an analytic structure (with separated power series) from [CL], hence inheriting all properties from loc. cit., including geometric properties for the definable sets like certain forms of quantifier elimination. Our expansion comes from adding specific, existentially definable functions, which are solutions of certain henselian systems of equations. Moreover, we show that, even when K is algebraically closed, one does not have quantifier elimination in \mathcal L_K itself, and hence, passing to expansions is unavoidable in general. We pursue this study in the wider generality of extending non-separated power series rings to separated ones, and give new examples, in particular of the analytic structure over \mathbb Z[[t]] that can be interpreted and understood now in all complete valued fields. In a separate direction, we show in rather large generality that Weierstrass preparation implies Weierstrass division.

Estimates for Weierstrass division in ultradifferentiable classes

We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy–Carleman classes CM. For suitable divisors P(x,t)=xd+a1(t)xd−1+⋯+ad(t) with real-analytic coefficients aj, we show that the quotient and the remainder can be chosen of class CMσ, where Mσ=((Mj)σ)j≥0 and σ is a certain Łojasiewicz exponent related to the geometry of the roots of P and verifying 1≤σ≤d. We provide various examples for which σ is optimal, in particular strictly less than d, which sharpens earlier results of Bronshtein and of Chaumat–Chollet.

Local zero estimates and effective division in rings of algebraic power series

Abstract We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally, we apply these results to prove a gap theorem for power series which are remainders of the Grauert–Hironaka–Galligo Division Theorem.

Integration of semialgebraic functions and integrated Nash functions

When integrating semialgebraic functions one has to leave the semialgebraic setting. For example, one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent. We show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions. To realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives. We analyze these rings profoundly. In particular we show that the Weierstrass division theorem and the Weierstrass preparation theorem hold. This allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions. Finally, we investigate the structure generated by the integrated algebraic power series.

Link between Noetherianity and the Weierstrass Division Theorem on some quasianalytic local rings

Link between Noetherianity and the Weierstrass Division Theorem on some quasianalytic local rings

Weierstrass preparation theorem for noncommutative rings

A power series over a complete local ring can be canonically decomposed into the product of an invertible power series and a unital polynomial whose degree coincides with the number of the first invertible coefficient. This statement is known as the Weierstrass preparation theorem. It follows from a more general statement known as the Weierstrass division theorem. The present article contains a detailed proof of generalizations of the Weierstrass preparation theorem and the Weierstrass division theorem for the so-called rings of skew power series. Such rings arise in number theory, and first in studies of formal groups over local fields. Bibliography: 3 titles.

Weierstrass division theorem in quasianalytic local rings

The main result of this paper is the following: if the Weierstrass division theorem is valid in a quasianalytic differentiable system, then this system is contained in the system of analytic germs. This result has already been known for particular example

On quasianalytic local rings

This expository article is devoted to the local theory of ultradifferentiable classes of functions, with a special emphasis on the quasianalytic case. Although quasianalytic classes are well-known in harmonic analysis since several decades, their study from the viewpoint of differential analysis and analytic geometry has begun much more recently and, to some extent, has earned them a new interest. Therefore, we focus on contemporary questions closely related to topics in local algebra. We study, in particular, Weierstrass division problems and the role of hyperbolicity, together with properties of ideals of quasianalytic germs. Incidentally, we also present a simplified proof of Carleman's theorem on the non-surjectivity of the Borel map in the quasianalytic case.

Weierstrass division theorem in definable C∞germs in a polynomially bounded o-minimal structure

We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of $\mathbb{R}^n$, of definable $C^{\infty}$ functions.

Weierstrass Division in Quasianalytic Local Rings

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).