Universality of high-strength tensors

TL;DR

This paper extends a theorem on the universality of high-strength tensors to polynomial functors, showing that specialization induces a quasi-order with unique minimal and maximal classes among dense orbits.

Contribution

It generalizes Kazhdan and Ziegler's theorem to polynomial functors using new techniques, revealing a structured quasi-order on elements with dense orbits.

Findings

Specialization induces a quasi-order on polynomial functor elements.

Existence of unique smallest and largest equivalence classes among dense orbits.

Extension of universality results to arbitrary polynomial functors.

Abstract

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order.