Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sard properties for polynomial maps in infinite dimension

Antonio Lerario, Luca Rizzi, Daniele Tiberio

TL;DR

This paper addresses the Sard problem for maps from infinite-dimensional domains to finite-dimensional targets, where classical Sard–Smale theory fails in general. It introduces a quantitative framework based on entropy dimension and Kolmogorov $n$-width to derive sharp Sard-type results for polynomial maps $f:H\to\mathbb{R}^m$ and identifies precise thresholds (via a semialgebraic constant $\beta_0(d,m)$) under which critical-value images have measure zero. A central contribution is the Sard criterion for well-approximated maps, which leverages finite-dimensional polynomial approximations $f_n$ and a controlled decay rate $q^{-n}$ to bound the entropy of critical-value images. The framework yields concrete applications to Endpoint maps of Carnot groups, proving Sard-type and strong-Sard statements for real-analytic and piecewise analytic control spaces and establishing surjectivity results on finite-dimensional control subspaces, thereby advancing the understanding of the sub-Riemannian Sard conjecture in several settings.

Abstract

Sard's theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true -- even under the assumption that the map is ``polynomial'' -- and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard's theorem in this setting. Our motivation comes from sub-Riemannian geometry and, as an application of our results, we prove the sub-Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece-wise real-analytic controls with large enough radius of convergence, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls.

Sard properties for polynomial maps in infinite dimension

TL;DR

This paper addresses the Sard problem for maps from infinite-dimensional domains to finite-dimensional targets, where classical Sard–Smale theory fails in general. It introduces a quantitative framework based on entropy dimension and Kolmogorov -width to derive sharp Sard-type results for polynomial maps and identifies precise thresholds (via a semialgebraic constant ) under which critical-value images have measure zero. A central contribution is the Sard criterion for well-approximated maps, which leverages finite-dimensional polynomial approximations and a controlled decay rate to bound the entropy of critical-value images. The framework yields concrete applications to Endpoint maps of Carnot groups, proving Sard-type and strong-Sard statements for real-analytic and piecewise analytic control spaces and establishing surjectivity results on finite-dimensional control subspaces, thereby advancing the understanding of the sub-Riemannian Sard conjecture in several settings.

Abstract

Sard's theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true -- even under the assumption that the map is ``polynomial'' -- and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard's theorem in this setting. Our motivation comes from sub-Riemannian geometry and, as an application of our results, we prove the sub-Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece-wise real-analytic controls with large enough radius of convergence, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls.
Paper Structure (27 sections, 42 theorems, 273 equations)

This paper contains 27 sections, 42 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. The Sard property for polynomial maps on infinite-dimensional spaces
  3. Quantitative semialgebraic geometry
  4. Applications to Endpoint maps of Carnot groups
  5. Closing thoughts on the Sard conjecture
  6. Acknowledgements
  7. Table of notations
  8. Quantitative variations estimates
  9. Semialgebraic sets and maps
  10. Dimension and stratifications
  11. Semialgebraic triviality
  12. Projections
  13. Thom-Milnor bound
  14. Definable choice
  15. Variations and their behaviour under polynomial maps
  16. ...and 12 more sections

Key Result

Theorem 1

Let $H$ be a Hilbert space and $K \subset H$ be a compact set such that, for some $q>1$, it holds Let $d,m \in \mathbb{N}$. There exists $\beta_0 = \beta_0(d,m)>1$ such that for all $f\in \mathscr{P}_{d}^m(H)$ and $\nu \leq m-1$ we have In particular, if $q> \beta_0$, then the Sard property holds on $K$:

Theorems & Definitions (93)

  • Definition 1
  • Theorem 1: Sard under $n$--width assumptions
  • Remark 2
  • Theorem 2: Counterexamples to Sard
  • Remark 3
  • Theorem 3: Sard on linear subspaces
  • Remark 4
  • Example 5
  • Theorem 4: Sard criterion for well-approximated maps
  • Theorem 5: Global Sard for special maps
  • ...and 83 more