Semialgebraic sections of semialgebraic $C^m$-jet bundles in dimensions $n \ge 3$ at fixed smoothness $m \ge 2$

Fefferman and Luli proved that over the plane, every semialgebraic bundle of $m$-jets (with respect to $C^m_{loc}(\mathbb{R}^2,\mathbb{R}^D)$) that admits a section also admits a semialgebraic section, and they explicitly left open the higher-dimensional analogue: for fixed $m \ge 2$ and $n \ge 3$, does a semialgebraic bundle $\mathcal{H}$ with respect to $C^m_{loc}(\mathbb{R}^n,\mathbb{R}^D)$ that has a section necessarily have a semialgebraic section, at exact smoothness, with no loss of derivatives and no prescribed modulus of continuity? This single bundle question simultaneously contains the semialgebraic Whitney extension problem raised by Bierstone and Milman and the fixed-smoothness semialgebraic solvability of linear equations with semialgebraic coefficients.

Background

The classical Whitney extension problem asks which functions $f : E \to \mathbb{R}$ on a closed subset $E \subset \mathbb{R}^n$ are restrictions of $C^m$ functions on $\mathbb{R}^n$ ($m$-times continuously differentiable with bounded derivatives); it was solved in full generality by Fefferman in 2006. In tame geometry one asks the analogous questions with all data and all solutions required to be semialgebraic: a semialgebraic set is a finite union of sets defined by finitely many polynomial equations and strict inequalities, and a semialgebraic function is a function whose graph is a semialgebraic set. Semialgebraic sets are the definable sets of the simplest o-minimal structure (an o-minimal expansion of the real field is a collection of 'definable' sets, closed under the usual first-order operations, in which every definable subset of $\mathbb{R}$ is a finite union of points and intervals), and most results below hold for definable sets in any fixed o-minimal structure.

Bierstone and Milman raised the semialgebraic Whitney problem (recorded in the problem list of the Second Workshop on Whitney Problems, 2009, and stated as Problem 1 in the source paper): given a semialgebraic function $\varphi : E \to \mathbb{R}$ that extends to some $C^m_{loc}$ function on $\mathbb{R}^n$ (that is, $m$-times continuously differentiable, no boundedness required), must $\varphi$ extend to a semialgebraic $C^m_{loc}$ function? Here $m$ is fixed, so one may not 'lose derivatives' — that is, settle for an extension of lower smoothness $C^{m'}$ with $m' < m$. A companion question (Problem 2) concerns a linear equation $A_1F_1 + \cdots + A_DF_D = f$ with semialgebraic coefficients on $\mathbb{R}^n$: if it has a $C^m_{loc}$ solution, must it have a semialgebraic one? This is the fixed-smoothness, semialgebraic-coefficient variant of the Brenner–Epstein–Hochster–Kollár problem, which asks the same for continuous solutions of equations with polynomial coefficients.

Fefferman and Luli unified these questions in the language of jets and bundles. For $F \in C^m_{loc}(\mathbb{R}^n,\mathbb{R}^D)$, the $m$-jet $J_xF$ is the Taylor polynomial of $F$ of degree $m$ at the point $x$; the vector space $\mathcal{P}$ of degree-$m$ polynomials in $n$ variables becomes a ring $\mathcal{R}_x$ under truncated multiplication $P \odot_x Q = J_x(PQ)$, the ring of $m$-jets at $x$. A bundle $\mathcal{H} = (\mathcal{H}(x))_{x \in E}$ over a subset $E \subset \mathbb{R}^n$ assigns to each $x \in E$ a fiber $\mathcal{H}(x) = f(x) + I(x) \subset \mathcal{R}_x^D$, a translate of a submodule of the free $\mathcal{R}_x$-module $\mathcal{R}_x^D$, with all fibers nonempty. A section of $\mathcal{H}$ is a function $F \in C^m_{loc}(\mathbb{R}^n,\mathbb{R}^D)$ with $J_xF \in \mathcal{H}(x)$ for all $x \in E$, and $\mathcal{H}$ is called semialgebraic if the total space $\{(x, P) : x \in E,\ P \in \mathcal{H}(x)\}$ is a semialgebraic set (this forces $E$ to be semialgebraic). Problems 1 and 2 arise as special cases: constraining the value of the jet at $x$ to equal $\varphi(x)$, respectively requiring the jets to satisfy the linear equation at $x$. The general Problem 3 asks: if a semialgebraic bundle with respect to $C^m_{loc}(\mathbb{R}^n,\mathbb{R}^D)$ has a section, must it have a semialgebraic section, i.e. a section whose graph is a semialgebraic set?

Prior status. The definable Whitney extension theorem — due to Kurdyka–Pawlucki and, independently, Thamrongthanyalak — states that a definable Whitney jet, meaning the full Taylor data given as a definable family satisfying the classical Whitney compatibility conditions — the conditions on a family of candidate Taylor polynomials that characterize exactly the families arising as the jets of a single $C^m$ function — on a definable closed set extends to a definable $C^m$ function; it does not answer the questions above because there the jet data are not given. Aschenbrenner and Thamrongthanyalak settled the $C^1$ case of the semialgebraic Whitney problem in all dimensions, using a definable version of Michael's continuous-selection theorem, which produces a continuous single-valued selection of a set-valued map (a rule assigning to each point a set of admissible values). Fefferman–Kollár settled the continuous Brenner–Epstein–Hochster–Kollár problem. Bierstone–Campesato–Milman proved, in all dimensions and o-minimal generality, a loss-of-derivatives version: there exists $r = r(m) > m$ such that if a system with semialgebraic data admits a $C^r$ solution, then it admits a semialgebraic $C^m$ solution. Parusiński–Rainer settled the $C^{1,\omega}$ case — first derivatives Hölder continuous with a prescribed definable modulus of continuity $\omega$ — in all dimensions for compact semialgebraic domains, via definable Lipschitz selections of affine-set-valued maps (set-valued maps whose values are affine subspaces), and extended the definable Whitney extension theorem to $C^{m,\omega}$-jets and definable families; Malagutti settled the continuous ($m = 0$) case of the linear-equation problem in all dimensions via Glaeser-stable bundles (the bundle obtained by iteratively discarding jets inconsistent with a Lipschitz-type bound). Fefferman and Luli proved Theorem 1 of the source paper: for $n = 2$, every semialgebraic bundle with respect to $C^m_{loc}(\mathbb{R}^2,\mathbb{R}^D)$ that has a section has a semialgebraic section, for every $m \ge 0$. Their proof localizes to thin plane wedges and reduces the problem to finitely many equations and inequalities for one-variable functions obtained as jet restrictions along semialgebraic curves — a reduction that is intrinsically two-dimensional. In the introduction they state: for $C^m_{loc}(\mathbb{R}^n)$ with $m \ge 2$ and $n \ge 3$, Problems 1, 2, 3 remain open.

Problem Statement

Fix integers $m \ge 2$ and $n \ge 3$, and $D \ge 1$. Let $\mathcal{H} = (\mathcal{H}(x))_{x \in E}$ be a semialgebraic bundle with respect to $C^m_{loc}(\mathbb{R}^n, \mathbb{R}^D)$: $E \subset \mathbb{R}^n$, each fiber is $\mathcal{H}(x) = f(x) + I(x) \subset \mathcal{R}_x^D$, a translate of an $\mathcal{R}_x$-submodule of the free module $\mathcal{R}_x^D$ over the ring $\mathcal{R}_x$ of degree-$m$ jets at $x$ (obtained from the vector space of real polynomials of degree at most $m$ in $n$ variables by truncated multiplication $P \odot_x Q = J_x(PQ)$, where $J_xG$ denotes the degree-$m$ Taylor polynomial of $G$ at $x$), all fibers are nonempty, and the total space $\{(x, P) : x \in E,\ P \in \mathcal{H}(x)\}$ is semialgebraic. Suppose $\mathcal{H}$ has a section, i.e. a function $F \in C^m_{loc}(\mathbb{R}^n, \mathbb{R}^D)$ — a map whose components are $m$-times continuously differentiable, with no boundedness requirement on the derivatives — whose jet satisfies $J_xF \in \mathcal{H}(x)$ for every $x \in E$. Must $\mathcal{H}$ have a semialgebraic section, i.e. a section whose graph $\{(x, F(x))\}$ is a semialgebraic subset of $\mathbb{R}^n \times \mathbb{R}^D$?

This is Problem 3 of Fefferman–Luli restricted to the parameter regime $m \ge 2$, $n \ge 3$ that their two-dimensional theorem left open, at exact smoothness — no loss of derivatives and no prescribed modulus of continuity. The question simultaneously contains its two instances in the same regime: Problem 1, the semialgebraic Whitney problem (a semialgebraic $\varphi : E \to \mathbb{R}$ that extends to a $C^m_{loc}$ function on $\mathbb{R}^n$ extends to a semialgebraic one), and Problem 2, semialgebraic solvability at fixed smoothness of linear equations $A_1F_1 + \cdots + A_DF_D = f$ with semialgebraic coefficients, together with their underdetermined systems. A complete affirmative answer establishes the implication for every $m \ge 2$ and $n \ge 3$; a single explicit semialgebraic bundle, with respect to $C^m_{loc}(\mathbb{R}^n, \mathbb{R}^D)$ for some $m \ge 2$ and $n \ge 3$, that admits a $C^m_{loc}$ section but provably no semialgebraic $C^m_{loc}$ section, refutes it. Partial advances — for example a proof for $n = 3$, or for a fixed $m \ge 2$ and all $n \ge 3$ — would constitute progress on the source question but not its full resolution.

References

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