arXiv subjects

José F. Fernando

Publications and source records attributed to José F. Fernando.

At least 19 recordsLinked to original sources

arXiv2026

Nash approximation of differentiable semialgebraic maps

Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $μ\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $μ$-approximation target space} (or a $({\mathcal N},μ)$-${\tt ats}$ for short) if it has the following universal approximation property: {\em For each $m\in{\mathbb N}$ and each locally compact semialgebraic subset $S\subset{\mathbb R}^m$, the subspace of Nash maps ${\mathcal N}(S,T)$ is dense in the space ${\mathcal S}^μ(S,T)$ of ${\mathcal C}^μ$ semialgebraic maps between $S$ and $T$}. A necessary condition to be a $({\mathcal N},μ)$-${\tt ats}$ is that $T$ is locally connected by analytic paths. In this paper we show: {\em Nash manifolds with corners are $({\mathcal N},μ)$-${\tt ats}$ for each $μ\geq0$}. As an application of a stronger version of the previous statement, we show that if two Nash maps $f,g:S\to Q$, where $S$ is a locally compact semialgebraic set of ${\mathbb R}^m$ and $Q$ is a Nash manifold with corners, are close enough in the (strong) Whitney's semialgebraic topology of ${\mathcal S}^0(S,T)$ (and consequently they are (continuous) semialgebraically homotopic), then $f,g$ are Nash homotopic.

math.AG↗

arXiv2026

Nash uniformization of chessboard sets by Nash manifolds with corners

Bierstone and Parusiński studied the desingularization of $d$-dimensional closed subanalytic sets and in particular of $d$-dimensional closed semialgebraic sets. Their main tools are Hironaka's desingularization of real algebraic sets (to `uniform' the Zariski closure of the closed semialgebraic set) and Hironaka's embedded desingularization of real algebraic subsets of non-singular real algebraic sets (to uniform afterwards the Zariski closure of the boundary of the uniformed closed semialgebraic set). The obtained models in the desingularization process, that we call in the following closed chessboard sets, are the closures of (finite) unions of connected components of the complements of normal-crossings divisors of non-singular real algebraic sets. The local models for $d$-dimensional chessboard sets are unions of (standard) closed orthants of ${\mathbb R}^d$, that is, $\bigcup_{(\varepsilon_1,\ldots,\varepsilon_d)\in{\mathfrak F}}\{\varepsilon_1{\tt x}_1\geq0,\ldots,\varepsilon_d{\tt x}_d\geq0\}\subset{\mathbb R}^d$ for some set ${\mathfrak F}\subset\{-1,1\}^d$. We study the Nash uniformization of $d$-dimensional closed chessboard sets ${\mathcal S}$ using Nash manifolds with corners ${\mathcal Q}$ with the same number of connected components as ${\mathcal S}$ (or equivalently the same number of irreducible components). Nash manifolds with corners are closed chessboard set whose local models are either ${\mathbb R}^d$ or semialgebraic sets of the type $\{{\tt x}_1\geq0,\ldots,{\tt x}_k\geq0\}$ for some $1\leq k\leq d$. More generally, a chessboard set is a semialgebraic set in between a finite union of connected components of the complement of a normal-crossings divisor of non-singular real algebraic set and its closure. We also provide a Nash uniformization result for general chessboard sets ${\mathcal S}$.

math.AG↗

arXiv2026

Applications of the Nash double of a Nash manifold with corners

In this work we study some properties and applications of Nash manifolds with corners. Our first main result shows how to `build' a Nash manifold with corners ${\mathcal Q}\subset{\mathbb R}^n$ from a suitable Nash manifold $M\subset{\mathbb R}^n$ (of its same dimension), that contains ${\mathcal Q}$ as a closed subset, by folding $M$ along the irreducible components of a normal-crossings divisor of $M$ (the smallest Nash subset of $M$ that contains the boundary $\partial{\mathcal Q}$ of ${\mathcal Q}$). Our second main results shows that we can choose as the Nash manifold $M$ the Nash `double' $D({\mathcal Q})$ of ${\mathcal Q}$, which is the analogous to the Nash double of a Nash manifold with (smooth) boundary, but $D({\mathcal Q})$ takes into account the peculiarities of the boundary of a Nash manifold with corners. We propose several applications of the previous results: (1) Nash ramified coverings of closed semialgebraic sets, (2) Weak Nash uniformization of closed semialgebraic sets using Nash manifolds with (smooth) boundary, (3) Representation of compact semialgebraic sets connected by analytic paths as images under Nash maps of closed unit balls, (4) Explicit construction of Nash models for compact orientable smooth surfaces of genus $g\geq0$, and (5) Nash approximation of continuous semialgebraic maps whose target spaces are Nash manifolds with corners.

math.AG↗

arXiv2025

Subfield-algebraic geometry

In this monograph, we lay the foundations for a new theory that generalizes real algebraic geometry. Let $R|K$ be a field extension, where $R$ is a real closed field and $K$ is an ordered subfield of $R$. The main objective is to study $K$-algebraic subsets of $R^n$, i.e., those subsets of $R^n$ that are the zero loci of polynomials with coefficients in $K$. Real algebraic geometry already covers the case when $K$ is also a real closed field. Our goal is to extend real algebraic geometry to the case when $K$ is not real closed, for example when $K$ is the field $\mathbb{Q}$ of rational numbers. Several new geometric phenomena appear. There is no complex counterpart to this generalized real algebraic geometry. The reason is as follows. If $C|K$ is a field extension with $C$ algebraically closed and $X$ is a $K$-algebraic subset of $C^n$, then Hilbert's Nullstellensatz implies that the ideal of polynomials with coefficients in $C$ that vanish on~$X$ is generated by the ideal of polynomials with coefficients in $K$ that vanish on $X$. In the real realm, this is false in general, for example when we consider field extensions $R|K$ with $R$ real closed and $K=\mathbb{Q}$. This monograph also presents some applications of the theory developed. Here is an example. The celebrated Nash-Tognoli theorem states that every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set $M'$, called algebraic model of $M$. The theory developed here provides the theoretical basis to prove that the algebraic model $M'$ of $M$ can be chosen to be $\mathbb{Q}$-algebraic and $\mathbb{Q}$-nonsingular. This guarantees for the first time that, up to smooth diffeomorphisms, every compact smooth manifold can be encoded both globally and locally involving only finitely many exact data.

math.AG↗

arXiv2025

A converse to Cartan's Theorem B: The extension property for real analytic and Nash sets

In 1957 Cartan proved his celebrated Theorem B and deduced that if $Ω\subset{\mathbb R}^n$ is an open set and $X$ is a coherent real analytic subset of $Ω$, then $X$ has the analytic extension property: Each real analytic function on $X$ extends to a real analytic function on $Ω$. The converse implication remains unproven. In the literature only special cases of non-coherent real analytic sets $X\subsetΩ$ without the extension property appear: mainly real analytic sets $X\subsetΩ$ that have a visible `tail'. We prove the converse implication: If $X\subsetΩ$ has the analytic extension property, it is a coherent real analytic subset of $Ω$. Taking advantage of cohomology of sheaves, we provide for each non-coherent real analytic set $X$, `many' failing analytic functions on $X$ (that have no analytic extension to $Ω$), yielding an almost complete description of the possible non-extendability sets. We extend the previous characterization to the Nash case, which is somehow more demanding, because of its finiteness properties and its disappointing behavior with respect to cohomology of sheaves of Nash function germs, and we provide again an almost complete description of the possible non-extendability sets. Let $Ω\subset{\mathbb R}^n$ be an open semialgebraic set and let $X\subsetΩ$: Each Nash function on $X$ extends to a Nash function on $Ω$ if and only if $X\subsetΩ$ is a coherent Nash set. The `if' implication goes back to some celebrated results of Coste, Ruiz and Shiota. If $M\subset{\mathbb R}^n$ is a Nash manifold, ${\mathcal C}^\infty$ semialgebraic functions on $M$ coincide with Nash functions on $M$. As an application of our results, we provide a full characterization of the semialgebraic sets $S\subsetΩ$ for which ${\mathcal C}^\infty$ semialgebraic functions on $S$ coincide with Nash functions on $S$.

math.AG↗

arXiv2025

On the one dimensional polynomial, regular and regulous images of closed balls and spheres

We present a full geometric characterization of the $1$-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls $\overline{\mathcal B}_n\subset{\mathbb R}^n$ or $n$-dimensional spheres ${\mathbb S}^n\subset{\mathbb R}^{n+1}$ under polynomial, regular and regulous maps for some $n\geq1$. In all the previous cases one can find an alternative polynomial, regular or regulous map on either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$ such that $S$ is the image under such map of either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$. As a byproduct, we provide a full characterization of the images of ${\mathbb S}^1\subset{\mathbb C}\equiv{\mathbb R}^2$ under Laurent polynomials $f\in{\mathbb C}[{\tt z},{\tt z}^{-1}]$, taking advantage of some previous works of Kobalev-Yang and Wilmshurst. We also alternatively prove that all polynomial maps ${\mathbb S}^k\to{\mathbb S}^1$ are constant if $k\geq2$.

math.AG↗

arXiv2025

On a Nash curve selection lemma through finitely many points

A celebrated theorem in Real Algebraic and Analytic Geometry (originally due to Bruhat-Cartan and Wallace and stated later in its current form by Milnor) is the (Nash) curve selection lemma. It states that each point in the closure of a semialgebraic set ${\mathcal S}\subset{\mathbb R}^n$ can be reached by a Nash arc of ${\mathbb R}^n$ such that at least one of its branches is contained in ${\mathcal S}$. The purpose of this work is to generalize the previous result to finitely many points. More precisely, let ${\mathcal S}\subset\R^n$ be a semialgebraic set, let $x_1,\ldots,x_r\in{\mathcal S}$ be $r$ points (that we call `control points') and $0=:t_1<\ldots<t_r:=1$ be $r$ values (that we call `control times'). A natural `logistic' question concerns the existence of a smooth and semialgebraic (Nash) path $α:[0,1]\to{\mathcal S}$ that passes through the control points at the control times, that is, $α(t_k)=x_k$ for $k=1,\ldots,r$. The necessary and sufficient condition to guarantee the existence of $α$ when the number of control points is large enough and they are in general position is that $\Ss$ is connected by analytic paths. The existence of generic real algebraic sets that do not contain rational curves confirms that the analogous result involving polynomial paths (instead of Nash paths) is only possible under additional restrictions. A sufficient condition is that $\Ss\subset\R^n$ has in addition dimension $n$. A related problem concerns the approximation by a Nash path of an existing continuous semialgebraic path $β:[0,1]\to{\mathcal S}$ with control points $x_1,\ldots,x_r\in\Ss$ and control times $0=:t_1<\ldots<t_r:=1$. A sufficient condition is that the (finite) set of values $η(β)$ at which $β$ is not smooth is contained in the set of regular points of ${\mathcal S}$ and $η(β)$ does not meet the set of control times.

math.AG↗

arXiv2024

Spectral Spaces in o-minimal and other NIP theories

We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compact set is a normal spectral space whose closed points are the finitely satisfiable types. On the other hand, for arbitrary NIP structures we equip the set of invariant types with a new topology, called the {\em honest topology}. With this topology the set of invariant types is a normal spectral space whose closed points are the finitely satisfiable ones, and the natural retraction from invariant types onto finitely satisfiable types coincides with Simon's $F_M$ retraction.

math.LO↗

arXiv2024

Positive semidefinite analytic functions on real analytic surfaces

Let $X\subset{\mathbb R}^n$ be a (global) real analytic surface. Then every positive semidefinite meromorphic function on $X$ is a sum of $10$ squares of meromorphic functions on $X$. As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces.

math.CV↗

arXiv2024

Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring $A$ as sums of squares of elements of $A$. If $A$ is an excellent ring of dimension $\geq3$, it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in $A$. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the $2$-dimensional case and determine (under some mild conditions) which local excellent henselian rings $A$ of embedding dimension $3$ have the property that every positive semidefinite element of $A$ is a sum of squares of elements of $A$.

math.AG↗

arXiv2024

Unbounded convex polyhedra as polynomial images of Euclidean spaces

In a previous work we proved that each $n$-dimensional convex polyhedron ${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images of ${\mathbb R}^n$. As the image of a non-constant polynomial map is an unbounded semialgebraic set, it is not possible to substitute regular maps by polynomial maps in the previous statement. In this work we determine constructively all unbounded $n$-dimensional convex polyhedra ${\mathcal K}\subset{\mathbb R}^n$ that are polynomial images of ${\mathbb R}^n$. We also analyze for which of them the interior ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$. A discriminating object is the recession cone $\vec{\mathcal C}({\mathcal K})$ of ${\mathcal K}$. Namely, \em ${\mathcal K}$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{\mathcal C}({\mathcal K})$ has dimension $n$\em. In addition, \em ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{\mathcal C}({\mathcal K})$ has dimension $n$ and ${\mathcal K}$ has no bounded faces of dimension $n-1$\em. A key result is an improvement of Pecker's elimination of inequalities to represent semialgebraic sets as projections of algebraic sets. Empirical approaches suggest us that there are `few' polynomial maps that have a concrete convex polyhedron as a polynomial image and that there are even fewer for which it is affordable to show that their images actually correspond to our given convex polyhedron. This search of a `needle in the haystack' justifies somehow the technicalities involved in our constructive proofs.

math.AG↗

arXiv2024

On polynomial images of a closed ball

In this work we approach the problem of determining which (compact) semialgebraic subsets of ${\mathbb R}^n$ are images under polynomial maps $f:{\mathbb R}^m\to{\mathbb R}^n$ of the closed unit ball $\overline{\mathcal B}_m$ centered at the origin of some Euclidean space ${\mathbb R}^m$ and that of estimating (when possible) which is the smallest $m$ with this property. Contrary to what happens with the images of ${\mathbb R}^m$ under polynomial maps, it is quite straightforward to provide basic examples of semialgebraic sets that are polynomial images of the closed unit ball. For instance, simplices, cylinders, hypercubes, elliptic, parabolic or hyperbolic segments (of dimension $n$) are polynomial images of the closed unit ball in ${\mathbb R}^n$. The previous examples (and other basic ones proposed in the article) provide a large family of `$n$-bricks' and we find necessary and sufficient conditions to guarantee that a finite union of `$n$-bricks' is again a polynomial image of the closed unit ball either of dimension $n$ or $n+1$. In this direction, we prove: {\em A finite union ${\mathcal S}$ of $n$-dimensional convex polyhedra is the image of the $n$-dimensional closed unit ball $\overline{\mathcal B}_n$ if and only if ${\mathcal S}$ is connected by analytic paths}. The previous result can be generalized using the `$n$-bricks' mentioned before and we show: {\em If ${\mathcal S}_1,\ldots,{\mathcal S}_\ell\subset{\mathbb R}^n$ are `$n$-bricks', the union ${\mathcal S}:=\bigcup_{i=1}^\ell{\mathcal S}_i$ is the image of the closed unit ball $\overline{\mathcal B}_{n+1}$ of ${\mathbb R}^{n+1}$ under a polynomial map $f:{\mathbb R}^{n+1}\to{\mathbb R}^n$ if and only if ${\mathcal S}$ is connected by analytic paths}.

math.AG↗

arXiv2023

Surjective Nash maps between semialgebraic sets

In this work we study the existence of surjective Nash maps between two given semialgebraic sets ${\mathcal S}$ and ${\mathcal T}$. Some key ingredients are: the irreducible components ${\mathcal S}_i^*$ of ${\mathcal S}$ (and their intersections), the analytic-path connected components ${\mathcal T}_j$ of ${\mathcal T}$ (and their intersections) and the relations between dimensions of the semialgebraic sets ${\mathcal S}_i^*$ and ${\mathcal T}_j$. A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps. The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: {\em a full characterization of the semialgebraic subsets ${\mathcal S}\subset{\mathbb R}^n$ that are the image of the closed unit ball $\overline{\mathcal B}_m$ of ${\mathbb R}^m$ centered at the origin under a Nash map $f:{\mathbb R}^m\to{\mathbb R}^n$}. The necessary and sufficient conditions that must satisfy such a semialgebraic set ${\mathcal S}$ are: {\em it is compact, connected by analytic paths and has dimension $d\leq m$}. Two remarkable consequences of the latter result are the following: (1) {\em pure dimensional compact irreducible arc-symmetric semialgebraic sets of dimension $d$ are Nash images of $\overline{\mathcal B}_d$}, and (2) {\em compact semialgebraic sets of dimension $d$ are projections of non-singular algebraic sets of dimension $d$ whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions)}.

math.AG↗

arXiv2021

Smooth approximations in PL geometry

Let $Y\subset{\mathbb R}^n$ be a triangulable set and let $r$ be either a positive integer or $r=\infty$. We say that $Y$ is a $\mathscr{C}^r$-approximation target space, or a $\mathscr{C}^r\text{-}\mathtt{ats}$ for short, if it has the following universal approximation property: For each $m\in{\mathbb N}$ and each locally compact subset $X$ of~${\mathbb R}^m$, any continuous map $f:X\to Y$ can be approximated by $\mathscr{C}^r$ maps $g:X\to Y$ with respect to the strong $\mathscr{C}^0$ Whitney topology. Taking advantage of new approximation techniques we prove: if $Y$ is weakly $\mathscr{C}^r$ triangulable, then $Y$ is a $\mathscr{C}^r\text{-}\mathtt{ats}$. This result applies to relevant classes of triangulable sets, namely: (1) every locally compact polyhedron is a $\mathscr{C}^\infty\text{-}\mathtt{ats}$, (2) every set that is locally $\mathscr{C}^r$ equivalent to a polyhedron is a $\mathscr{C}^r\text{-}\mathtt{ats}$, and (3) every locally compact locally definable set of an arbitrary o-minimal structure is a $\mathscr{C}^1\text{-}\mathtt{ats}$ (this includes locally compact locally semialgebraic sets and locally compact subanalytic sets). In addition, we prove: if $Y$ is a global analytic set, then each proper continuous map $f:X\to Y$ can be approximated by proper $\mathscr{C}^\infty$ maps $g:X\to Y$. Explicit examples show the sharpness of our results.

math.DG↗

arXiv2019

Rings of differentiable semialgebraic functions

In this work we analyze the main properties of the Zariski and maximal spectra of the ring ${\mathcal S}^r(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^r$ on a semialgebraic set $M\subset\mathbb{R}^m$. Denote ${\mathcal S}^0(M)$ the ring of semialgebraic functions on $M$ that admit a continuous extension to an open semialgebraic neighborhood of $M$ in $\text{cl}(M)$. This ring is the real closure of ${\mathcal S}^r(M)$. If $M$ is locally compact, the ring ${\mathcal S}^r(M)$ enjoys a Lojasiewicz's Nullstellensatz, which becomes a crucial tool. Despite ${\mathcal S}^r(M)$ is not real closed for $r\geq1$, the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring ${\mathcal S}^0(M)$. In addition, the quotients of ${\mathcal S}^r(M)$ by its prime ideals have real closed fields of fractions, so the ring ${\mathcal S}^r(M)$ is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of ${\mathcal S}^r(M)$ and ${\mathcal S}^0(M)$ guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring ${\mathcal S}^r(M)$ is a Gelfand ring and its Krull dimension is equal to $\dim(M)$. We also show similar properties for the ring ${\mathcal S}^{r*}(M)$ of differentiable bounded semialgebraic functions. In addition, we confront the ring ${\mathcal S}^{\infty}(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^{\infty}$ with the ring ${\mathcal N}(M)$ of Nash functions on $M$.

math.AG↗

arXiv2018

Differentiable approximation of continuous semialgebraic maps

In this work we approach the problem of approximating uniformly continuous semialgebraic maps $f:S\to T$ from a compact semialgebraic set $S$ to an arbitrary semialgebraic set $T$ by semialgebraic maps $g:S\to T$ that are differentiable of class~${\mathcal C}^ν$ for a fixed integer $ν\geq1$. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For $ν=1$ we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For $ν\geq 2$ we obtain density results in the two following relevant situations: either $T$ is compact and locally ${\mathcal C}^ν$ semialgebraically equivalent to a polyhedron, for instance when $T$ is a compact polyhedron; or $T$ is an open semialgebraic subset of a Nash set, for instance when $T$ is a Nash set. Our density results are based on a recent ${\mathcal C}^1$-triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.

math.AG↗

arXiv2018

On Nash images of Euclidean spaces

In this work we characterize the subsets of ${\mathbb R}^n$ that are images of Nash maps $f:{\mathbb R}^m\to{\mathbb R}^n$. We prove Shiota's conjecture and show that a subset ${\mathcal S}\subset{\mathbb R}^n$ is the image of a Nash map $f:{\mathbb R}^m\to{\mathbb R}^n$ if and only if ${\mathcal S}$ is semialgebraic, pure dimensional of dimension $d\leq m$ and there exists an analytic path $α:[0,1]\to{\mathcal S}$ whose image meets all the connected components of the set of regular points of ${\mathcal S}$. Some remarkable consequences are the following: (1) pure dimensional irreducible semialgebraic sets of dimension $d$ with arc-symmetric closure are Nash images of ${\mathbb R}^d$; (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces; and (3) compact $d$-dimensional smooth manifolds with boundary are smooth images of ${\mathbb R}^d$.

math.AG↗

arXiv2018

On the set of local extrema of a subanalytic function

Let ${\mathfrak F}$ be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical and basic topological operations. Let $M$ be a real analytic manifold and denote ${\mathfrak F}(M)$ the family of the subsets of $M$ that belong to ${\mathfrak F}$. Let $f:X\to{\mathbb R}$ be a subanalytic function on a subset $X\in{\mathfrak F}(M)$ such that the inverse image under $f$ of each interval of ${\mathbb R}$ belongs to ${\mathfrak F}(M)$. Let ${\rm Max}(f)$ be the set of local maxima of $f$ and consider ${\rm Max}_λ(f):={\rm Max}(f)\cap\{f=λ\}$ for each $λ\in{\mathbb R}$. If $f$ is continuous, then ${\rm Max}(f)=\bigsqcup_{λ\in{\mathbb R}}{\rm Max}_λ(f)\in{\mathfrak F}(M)$ if and only if the family $\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$ is locally finite in $M$. If we erase continuity condition, there exist subanalytic functions $f:X\to M$ such that ${\rm Max}(f)\in{\mathfrak F}(M)$, but the family $\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$ is not locally finite in $M$ or such that ${\rm Max}(f)$ is connected but it is not even subanalytic. If ${\mathfrak F}$ is the category of subanalytic sets and $f:X\to{\mathbb R}$ is a subanalytic map $f$ that maps relatively compact subsets of $M$ contained in $X$ to bounded subsets of ${\mathbb R}$, then ${\rm Max}(f)\in{\mathfrak F}(M)$ and the family $\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$ is locally finite in $M$. If the category ${\mathfrak F}$ contains the intersections of algebraic sets with real analytic submanifolds and $X\in{\mathfrak F}(M)$ is not closed in $M$, there exists a continuous subanalytic function $f:X\to{\mathbb R}$ with graph belonging to ${\mathfrak F}(M\times{\mathbb R})$ such that inverse images under $f$ of the intervals of ${\mathbb R}$ belong to ${\mathfrak F}(M)$ but ${\rm Max}(f)$ does not belong to ${\mathfrak F}(M)$.

math.AG↗