Let X be a real algebraic set. In the literature, a (real-valued) function φ on X is called regulous if its restriction to each algebraic subset of X is a continuous rational function. A function f on X is called quasi-regulous if it is continuous and f^2=φ ^2 for some regulous function φ on X. Assuming X is nonsingular, we prove that a function f on X is quasi-regulous if and only if its restriction to each algebraic curve in X is quasi-regulous. We also prove three other variants of this result.

Let $X$ be a quasi-projective algebraic variety over a real closed field $R$, and let $f \colon U \to R$ be a function defined on an open subset $U$ of the set $X(R)$ of $R$-rational points of $X$. Assume that either the function $f$ is locally semialgebraic or the field $R$ is uncountable. If for every irreducible algebraic curve $C \subset X$ the restriction $f|_{U \cap C}$ is continuous and admits a rational representation, then $f$ is continuous and admits a rational representation. There are also suitable versions of this theorem with algebraic curves replaced by algebraic arcs. Heretofore, results of such a type have been known only for $R={\mathbb{R}}$. The transition from ${\mathbb{R}}$ to $R$ is not automatic at all and requires new methods.

Let X be a real analytic manifold. A function f :X →ℝ is said to be curve-analytic if it is real analytic when restricted to any locally irreducible real analytic curve in X . We prove that every curve-analytic function with subanalytic graph is actually real analytic. To accomplish this task, we give a criterion for an arc-analytic function to be real analytic. A function is called arc-analytic if it is real analytic along any parametric real analytic arc. We also obtain analogous results for Nash manifolds and Nash functions, in which case the assumption of subanalyticity is superfluous.

Let R be a real closed field. We prove that if R is uncountable, then a function f: U → R defined on an open semialgebraic set U in Rn, with n ≥ 2, is a Nash function whenever for every affine 2-plane Q in Rnthe restriction f∣U∩Q is a Nash function (some condition on the shape of U is required if R is not Archimedean). This is an analog of the Bochnak-Siciak theorem established in the real analytic setting. We also provide an example showing that uncountability of R is essential.

Let X⊂Rn be a convex closed and semialgebraic set and let f be a polynomial positive on X. We prove that there exists an exponent N⩾1, such that for any ξ∈Rn the function φN(x)=eN|x−ξ|2f(x) is strongly convex on X. When X is unbounded we have to assume also that the leading form of f is positive in Rn∖{0}. We obtain strong convexity of ΦN(x)=eeN|x|2f(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.

We propose a method to bound the length of gradient trajectories by comparison with the length of corresponding talwegs (ridge or valley lines), and we obtain several applications. We show that gradient trajectories of a definable (in an o-minimal structure) family of functions are of uniformly bounded length. We prove that the length of a trajectory of the gradient of a polynomial in n variables of degree d in a ball of radius r is bounded by rA(n, d), where A(n, d) = nu(n)((3d - 4)(n-1) + 2(3d - 3)(n-2)) and nu(n) is an explicit constant. We give explicit bounds for the length of gradient trajectories of quasipolynomials and trigonometric quasipolynomials. As an application we give a construction of a curve (piecewise gradient trajectory of a polynomial) joining two points in an open connected semialgebraic set. We give an explicit bound for its length. We also obtain an explicit and quite sharp bound in Yomdin's version of the quantitative Morse-Sard theorem.

Let X be an irreducible smooth real algebraic variety of dimension at least 2 and let f : U → R be a function defined on a connected open subset U ⊂ R). Assume that for every irreducible smooth real algebraic curve C ⊂ X, for which C(R) is the boundary of a disc embedded in U , the restriction f |C(R) is continuous and has a rational representation. Then f has a rational representation. This is a significant refinement of a recent result of J. Kollár and the authors. The novelty is that existence of rational representation is tested on a much smaller and more rigid class of curves. We also consider the case where U is not necessarily connected and test rationality on subvarieties of dimension greater than 1. For semialgebraic functions our results hold under slightly weaker assumptions. Joint work with K. Kurdyka

Let R be a real closed field. We prove that if R is uncountable, then any separately Nash (respectively, arc-Nash) function defined over R is semialgebraic (respectively, continuous semialgebraic). To complete the picture, we provide an example showing that the assumption on R to be uncountable cannot be dropped. Moreover, even if R is uncountable but non-Archimedean, then the shape of the domain of a separately Nash function matters for the conclusion. For R=R, we prove that arc-Nash functions coincide with arc-analytic semialgebraic functions.

Let R be a real closed field. We prove that if R is uncountable, then any separately Nash (respectively, arc-Nash) function defined over R is semialgebraic (respectively, continuous semialgebraic). To complete the picture, we provide an example showing that the assumption on R to be uncountable cannot be dropped. Moreover, even if R is uncountable but non-Archimedean, then the shape of the domain of a separately Nash function matters for the conclusion. For R = R , we prove that arc-Nash functions coincide with arc-analytic semialgebraic functions.

We introduce arc-meromorphous functions, which are continuous functions representable as quotients of semialgebraic arc-analytic functions, and develop the theory of arc-meromorphous sheaves on Nash manifolds. Our main results are Cartan's theorems A and B for quasi-coherent arc-meromorphous sheaves.

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.

Let f be a semialgebraic function of class C-1, defined on an open set U subset of R-n. Let P(x,y)is an element of R[x(1),( )..., x(n), y] be a polynomial of degree d such that P(x, f (x)) = 0, x is an element of U. We prove that if f is of class C-K with K > 1/2d(7), then f is analytic. If n = 1, then it suffices that K > 1/2d(2).

We give various quantitative versions of Łojasiewicz inequalities for semialgebraic sets and mappings, both in the local and global case.

Gdy f jest funkcją analityczną przełomowe odkrycie S. Łojasiewicza [1] nierówności ∇f ≥ c|f |ρ , z wykładnikiem ρ < 1, pozwala udowodnić, że długość trajektorii ∇f między dwoma poziomicami f jest skończona. Wynika stąd istnienie granicy dowolnej trajektorii ∇f . W [2] podałem jej uogólnienie dla szerokiej klasy funkcji definiowalnych w strukturach o-minimalnych. Pod nazwą nierówności (warunku) K-L jest stosowana do wykazywania zbieżności algorytmów w teorii optymizacji. Ogólniejsza metoda oszacowania długości trajektorii ∇f przez długość talwegu (linii dna doliny) ma potencjalnie jeszcze szersze pole zastosowań, np. [3]. W [4] wykazaliśmy istnienie granicy siecznych w punkcie krytycznym trajektorii gradientu funkcji analitycznej f . Bardziej subtelny opis trajektorii w pobliżu punktów krytycznych f pozostaje dalej otwarty.

We give a bound of the height of a multipolynomial resultant in terms of polynomial degrees, the resultant of which applies.Additionally we give a Gelfond-Mahler type bound of the height of homogeneous divisors of a homogeneous polynomial.

Proceedings of the International Congress of Mathematicians (ICM 2018), pp. 719-747 (2019) No AccessFROM CONTINUOUS RATIONAL TO REGULOUS FUNCTIONSWOJCIECH KUCHARZ and KRZYSZTOF KURDYKAWOJCIECH KUCHARZINSTITUTE OF MATHEMATICS, FACULTY OF MATHEMATICS AND COMPUTER SCIENCE, JAGIELLONIAN UNIVERSITY, ŁOJASIEWICZA 6, 30-348 KRAKÓW, POLAND and KRZYSZTOF KURDYKAUNIV. GRENOBLE ALPES, UNIV. SAVOIE MONT BLANC, CNRS, LAMA. 73000 CHAMBÉRY, FRANCEhttps://doi.org/10.1142/9789813272880_0075Cited by:1 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: Let X be an algebraic set in ℝn. Real-valued functions, defined on subsets of X, that are continuous and admit a rational representation have some remarkable properties and applications. We discuss recently obtained results on such functions, against the backdrop of previously developed theories of arc-symmetric sets, arc-analytic functions, approximation by regular maps, and algebraic vector bundles. Keywords: Real algebraic setsemialgebraic setregular functionrational functionregulous functionarc-symmetric setarc-analytic functionapproximationvector bundleMSC2010: primary 14P05, secondary 14P99, secondary 26C15, secondary 57R22, secondary 58A07 FiguresReferencesRelatedDetailsCited By 1Separately Nash and arc‐Nash functions over real closed fieldsWojciech Kucharz, Krzysztof Kurdyka and Ali El‐Siblani9 November 2020 | Bulletin of the London Mathematical Society, Vol. 53, No. 2 Proceedings of the International Congress of Mathematicians (ICM 2018)Metrics History KeywordsReal algebraic setsemialgebraic setregular functionrational functionregulous functionarc-symmetric setarc-analytic functionapproximationvector bundlePDF download

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.

In this paper, we prove a version of global \L ojasiewicz inequality for $C^1$ semialgebraic functions and relate its existence to the set of asymptotic critical values.