The set of forms with bounded strength is not closed

Polynomials and tensors of bounded strength

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength. This generalises a theorem by Derksen-Eggermont-Snowden on cubic polynomials, as well as a theorem by Kazhdan-Ziegler which says that a polynomial all of whose directional derivati…

Lipschitz stratifications in power-bounded o-minimal fields

We propose to grok Lipschitz stratifications from a non-archimedean point of view and thereby show that they exist for closed definable sets in any power-bounded o-minimal structure on a real closed field. Unlike the previous approaches in the literature, our method bypasses resolution of singularities and Weierstrass preparation altogether; it transfers the situation to a non-archimedean model, where the quantitative estimates appearing in Lipschitz stratifications are sharpened into valuation-theoretic inequalities. Applied to a uniform family of sets, this approach automatically yields a fa…

$L^2$-extension of $\dbar$-closed forms

Generalizing and strengthening a recent result of Koziarz, we prove a version of the Ohsawa-Takegoshi-Manivel theorem for $\dbar$-closed forms.

强度由 Birch 秩线性界定

Strength is bounded linearly by Birch rank

本文证明数域与函数域上齐次多项式的强度可由 Birch 秩线性界定，并推出圆法在更弱算术条件下成立，同时改进有限域张量分区秩界。