Sard properties for polynomial maps in infinite dimension

TL;DR

This paper establishes precise conditions under which Sard's theorem holds for polynomial maps from infinite-dimensional Banach spaces to finite-dimensional spaces, with applications to sub-Riemannian geometry.

Contribution

It provides sharp quantitative criteria for Sard's theorem validity in infinite-dimensional polynomial maps, addressing a gap in the theory and applying it to sub-Riemannian geometry.

Findings

Proved Sard's theorem for polynomial maps under new sharp criteria.

Validated the sub-Riemannian Sard conjecture for Carnot groups.

Extended Sard's theorem applicability to infinite-dimensional polynomial maps.

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.