Extensions of regular mappings and the ?ojasiewicz exponent at infinity

Let F:V→Cm be a regular mapping, where V⊂Cn is an algebraic set of positive dimension and m?n?2, and let L_∞(F) be the ?ojasiewicz exponent at infinity of F. We prove that F has a polynomial extension G:Cn→Cm such L_∞(G)=L_∞(F). Moreover, we give an estimate of the degree of the extension G. Additionally, we prove that if then for any β∈Q, β?L_∞(F), the mapping F has a polynomial extension G with L_∞(G)=β. We also give an estimate of the degree of this extension.     

Injectivity of real polynomial maps and Łojasiewicz exponents at infinity

In this paper we give a lower bound for the Łojasiewicz exponent at infinity of a special class of polynomial maps , s ≥ 1. As a consequence, we detect a class of polynomial maps that are global diffeomorphisms if their Jacobian determinant never vanishes. Work supported by DGICYT Grant BFM2003–02037/MATE.     

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

Łojasiewicz exponents and resolution of singularities

We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of O_X{\mathcal{O}_X} , where X is a non-singular scheme. This method is based on the algorithm of resolution of singularities.     

Multiplicities of Pfaffian intersections,and the Łojasiewicz inequality

An effective estimate for the local multiplicity of a complete intersection of complex algebraic and Pfaffian varieties is given, based on a local complex analog of the Rolle-Khovanskii theorem. The estimate is valid also for the properly defined multiplicity of a non-isolated intersection. It implies, in particular, effective estimates for the exponents of the polar curves, and the exponents in the ojasiewicz inequalities for Pfaffian functions. For the intersections defined by sparse polynomials, the multiplicities outside the coordinate hyperplanes can be estimated in terms of the number of non-zero monomials, independent of degrees of the monomials.     

Semialgebraic sets and the Łojasiewicz-Siciak condition

The paper contains a full geometric characterization of compact semialgebraic sets in C satisfying the ?ojasiewicz-Siciak condition. The ?ojasiewicz-Siciak condition is a certain estimate for the Siciak extremal function. In a previous paper, we gave a sufficient criterion for a compact, connected, and semialgebraic set in C to satisfy this condition. In the present paper, we remove completely the connectedness assumption and prove that the aforementioned sufficient condition is also necessary. Moreover, we obtain some new results concerning the ?ojasiewicz-Siciak condition in C^N. For example, we prove that if K₁,...,K_p are compact, nonpluripolar, and pairwise disjoint subsets of C^N, each satisfying the ?ojasiewicz-Siciak condition, and K:= K₁?· · ·?K_p is polynomially convex, then K satisfies this condition as well.     

Effective formulas for the local ?ojasiewicz exponent

We give an effective formula for the local ?ojasiewicz exponent of a polynomial mapping. Moreover, we give an algorithm for computing the local dimension of an algebraic variety.     

The ?ojasiewicz exponent of nondegenerate surface singularities

We give some estimations of the ?ojasiewicz exponent of nondegenerate surface singularities in terms of their Newton diagrams. We also give an exact formula for the ?ojasiewicz exponent of such singularities in some special cases. The results are stronger than Fukui inequality?[8]. It is also a multidimensional generalization of the Lenarcik theorem?[13].     

Local Łojasiewicz exponents,Milnor numbers and mixed multiplicities of ideals

Let be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I ₁,..., I _n . We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions and Newton non-degenerate functions. Work supported by DGICYT Grant MTM2006-06027.     

The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

The aim of this paper is to address a problem raised originally by L. Gendre, later by W. Ple?niak and recently by L. Bia?as–Cie? and M. Kosek. This problem concerns the pluricomplex Green function and consists in finding new examples of sets with so–called ?ojasiewicz–Siciak ((?S) for short) property. So far, the known examples of such sets are rather of particular nature. We prove that each compact subset of ? ^N , treated as a subset of ? ^N , satisfies the ?ojasiewicz–Siciak condition. We also give a sufficient geometric criterion for a semialgebraic set in ?², but treated as a subset of ?, to satisfy this condition. This criterion applies more generally to a set in ? definable in a polynomially bounded o–minimal structure.     

An algebraic construction of the ring of¶piecewise polynomial functions

In this note we will prove an algebraic characterization of the piecewiese polynomial functions of a real ℚ-algebra A. This characterization is related to the realm of investigations concerning the Pierce–Birkhoff conjecture. Received: 23 June 1997 / Revised version: 26 May 1998     

Effective formulas for the ?ojasiewicz exponent at infinity

We give effective formulas for the ?ojasiewicz exponent at infinity of an arbitrary complex polynomial mapping.     

Convergence of Non-smooth Descent Methods Using the Kurdyka–Łojasiewicz Inequality

We investigate the convergence of subgradient-oriented descent methods in non-smooth non-convex optimization. We prove convergence in the sense of subsequences for functions with a strict standard model, and we show that convergence to a single critical point may be guaranteed if the Kurdyka–?ojasiewicz inequality is satisfied. We show, by way of an example, that the Kurdyka–?ojasiewicz inequality alone is not sufficient to prove the convergence to critical points.     

Virtual Poincaré polynomial of the link of a real algebraic variety

The link of an affine real algebraic variety at a point is defined to be the intersection of the variety with a small sphere centred at the point. The Euler characteristic of the link leads to local topological characterisation of real algebraic varieties in law dimensions. We prove in the paper that not only the Euler characteristic, but also the stronger virtual Poincaré polynomial is well-defined for the link at a point of an affine algebraic variety.     

Complete orthonormal sets of polynomial solutions of the Riesz and Moisil-Teodorescu systems in ℝ3

As it is well-known, the generalization of the classical Cauchy-Riemann system to higher dimensions leads to the so-called Riesz and Moisil-Teodorescu systems. Rewriting these systems in quaternionic language and taking advantage of the underlying algebra, we construct complete sets of polynomials solutions of both systems that are orthonormal with respect to a certain inner product. The restrictions of those polynomials to the unit sphere can be viewed as analogues to the complex case of the Fourier exponential functions $\{e^{i n \theta}\}_{n\geq 0}$ on the unit circle and constitute a refinement of the well-known spherical harmonics.     

The ?ojasiewicz exponent of a set of weighted homogeneous ideals

We give an expression for the ?ojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the ?ojasiewicz exponent of the gradient of a semi-weighted homogeneous function (Cⁿ,0)→(C,0) with an isolated singularity at the origin.     

Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods

In this paper, we study the Kurdyka–?ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is \(\frac{1}{2}\). The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is \(\frac{1}{2}\). This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with \(\ell _1\) regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery.