2023/08/28 by Gil Goldman, Goldman, Gil, Yosef Yomdin +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results
paper · pdf · doi:10.48550/arxiv.2308.14722
openalex publication_date 2023/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f: Bn → \mathbb R be a d+1 times continuously differentiable function on the unit ball Bn, with maxz∈ Bn ‖f(z)‖=1. A well-known fact is that if f vanishes on a set Z⊂ Bn with a non-empty interior, then for each k=1,…,d+1 the norm of the k-th derivative ‖f(k)‖ is at least M=M(n,k)>0. A natural question to ask is ``what happens for other sets Z?''. This question was partially answered in [16]-[18]. In the present paper we ask for a similar (and closely related) question: what happens with the high-order derivatives of f, if its gradient vanishes on a given set Σ? And what conclusions for the high-order derivatives of f can be obtained from the analysis of the metric geometry of the ``critical values set'' f(Σ)? In the present paper we provide some initial answers to these questions.