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Krzysztof Kurdyka

Publications and source records attributed to Krzysztof Kurdyka.

At least 19 recordsLinked to original sources
arXiv2018

Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

math.AG↗
arXiv2018

Exponential convexifying of polynomials

Let $X\subset\mathbb{R}^n$ be a convex closed and semialgebraic set and let $f$ be a polynomial positive on $X$. We prove that there exists an exponent $N\geq 1$, such that for any $ξ\in\mathbb{R}^n$ the function $φ_N(x)=e^{N|x-ξ|^2}f(x)$ is strongly convex on $X$. When $X$ is unbounded we have to assume also that the leading form of $f$ is positive in $\mathbb{R}^n\setminus\{0\}$. We obtain strong convexity of $\varPhi_N(x)=e^{e^{N|x|^2}}f(x)$ on possibly unbounded $X$, provided $N$ is sufficiently large, assuming only that $f$ is positive on $X$. We apply these results for searching critical points of polynomials on convex closed semialgebraic sets.

math.AG↗
arXiv2017

Higher order approximation of analytic sets by topologically equivalent algebraic sets

It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.

math.CV↗
arXiv2017

Tangent cones and $C^1$ regularity of definable sets

Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear subspace of $\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in $x$, and the density $θ(X, x) < 3/2$.

math.GT↗
arXiv2017

Curve-rational functions

Let $W$ be a subset of the set of real points of a real algebraic variety $X$. We investigate which functions $f: W \to \mathbb R$ are the restrictions of rational functions on $X$. We introduce two new notions: ${\it curve-rational \, functions}$ (i.e., continuous rational on algebraic curves) and ${\it arc-rational\, functions}$ (i.e., continuous rational on arcs of algebraic curves). We prove that under mild assumptions the following classes of functions coincide: continuous hereditarily rational (introduced recently by the first named author), curve-rational and arc-rational. In particular, if $W$ is semialgebraic and $f$ is arc-rational, then $f$ is continuous and semialgebraic. We also show that an arc-rational function defined on an open set is arc-analytic (i.e., analytic on analytic arcs). Furthermore, we study rational functions on products of varieties. As an application we obtain a characterization of regular functions. Finally, we get analogous results in the framework of complex algebraic varieties.

math.AG↗
arXiv2016

Linear equations on real algebraic surfaces

We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher dimensions.

math.AG↗
arXiv2015

Some conjectures on continuous rational maps into spheres

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results.

math.AG↗
arXiv2015

Comparison of stratified-algebraic and topological K-theory

Stratified-algebraic vector bundles on real algebraic varieties have many desirable features of algebraic vector bundles but are more flexible. We give a characterization of the compact real algebraic varieties having the following property: There exists a positive integer r such that for any topological vector bundle E on X, the direct sum of r copies of E is isomorphic to a stratified-algebraic vector bundle. In particular, each compact real algebraic variety of dimension at most 8 has this property. Our results are expressed in terms of K-theory.

math.AG↗
arXiv2015

Convexifying positive polynomials and sums of squares approximation

We show that if a polynomial $f\in \mathbb{R}[x_1,\ldots,x_n]$ is nonnegative on a closed basic semialgebraic set $X=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\ldots,g_r (x)\ge 0\}$, where $g_1,\ldots,g_r\in\mathbb{R}[x_1,\ldots,x_n]$, then $f$ can be approximated uniformly on compact sets by polynomials of the form $σ_0+φ(g_1) g_1+\cdots +φ(g_r) g_r$, where $σ_0\in \mathbb{R}[x_1,\ldots,x_n]$ and $φ\in\mathbb{R}[t]$ are sums of squares of polynomials. In particular, if $X$ is compact, and $h(x):=R^2-|x|^2 $ is positive on $X$, then $f=σ_{0}+σ_1 h+φ(g_1) g_1+\cdots +φ(g_r) g_r$ for some sums of squares $σ_{0},σ_1\in \mathbb{R}[x_1,\ldots,x_n]$ and $φ\in\mathbb{R}[t]$, where $|x|^2={x_1^2+\cdots+x_n^2}$. We apply a quantitative version of those results to semidefinite optimization methods. Let $X$ be a convex closed semialgebraic subset of $\mathbb{R}^n$ and let $f$ be a polynomial which is positive on $X$. We give necessary and sufficient conditions for the existence of an exponent $N\in\mathbb{N}$ such that $(1+|x|^2)^Nf(x)$ is a convex function on $X$. We apply this result to searching for lower critical points of polynomials on convex compact semialgebraic sets.

math.AG↗
arXiv2015

Nuij type pencils of hyperbolic polynomials

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations. In fact we show that all these sequences come from special symmetric Toeplitz matrices.

math.CA↗
arXiv2014

Metric properties of semialgebraic mappings

We give an effective estimation from above for the local Łojasiewicz exponent for separation of semialgebraic sets and for a semialgebraic mapping on a closed semialgebraic set. We also give an effective estimation from below of the Łojasiewicz exponent in the global separation for semialgebraic sets and estimation of the Łojasiewicz exponent at infinity of a semialgebraic mapping similar to the Jelonek result in the complex case.

math.AG↗
arXiv2014

Horizontal gradient of polynomial functions for the standard Engel structure on $\mathbb R^4$

We investigate the set $V_f$ of horizontal critical points of a polynomial function $f$ for the standard Engel structure defined by the 1-forms $ω_3=dx_3-x_1dx_2,$ $ω_4=dx_4-x_3dx_2$, endowed with the sub-Riemannian metric $g_{SR}=dx_1^2+dx_2^2$. For a generic polynomial, we show that the intersection of any fiber of $f$ and $V_f$ does not contain a horizontal curve. Then we prove that each trajectory of the horizontal gradient of $f$ approaching the set $V_f$ has a limit.

math.DS↗
arXiv2013

Stratified-algebraic vector bundles

We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification of X such that the restriction of the bundle to each stratum is an algebraic vector bundle. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which distinguish them from algebraic and topological vector bundles.

math.AG↗
arXiv2013

Reaching generalized critical values of a polynomial

Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.

math.AG↗
arXiv2010

Inverse Function Theorems for Arc-analytic Homeomorphisms

We call a local homeomorphism $f: (R^n,0)\to(R^n,0)$ blow-analytic if it becomes real analytic after composing with a finite number blowings-up with smooth nowhere dense centers. If the graph of $f$ is semi-algebraic then, by a theorem of Bierstone and Milman, $f$ is blow-analytic if and only if it is arc-analytic: the image by $f$ of a parametrized real analytic arc is again a real analytic arc. For a semialgebraic homeomorphism $f$ we show that if $f$ is blow-analytic and the inverse of $f$ is Lipschitz, then $f$ is Lipschitz and the inverse of $f$ is blow-analytic. The proof is by a motivic integration argument, using additive invariants on the spaces of arcs.

math.AG↗
arXiv2009

On the non-analyticity locus of an arc-analytic function

In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. Recall that a function is called arc-analytic if it is real analytic on each real analytic arc. By a result of Bierstone and Milman a big class of arc-analytic function, namely those that satisfy a polynomial equation with real analytic coefficients, can be made analytic by a sequence of global blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity locus.

math.AG↗
arXiv2007

Tame nonsmooth inverse mapping theorems

We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity of some principal minors, contractiblity) of the space of Jacobi's matrices at smooth points.

math.GT↗