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.
Highlights
Let K be an algebraically closed field of characteristic 0
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
Summary
Let K be an algebraically closed field of characteristic 0. Main Theorem I Let P be a property that, for each finite-dimensional vector space V , can be satisfied by some elements of Sλ(V ). Either P is satisfied by all elements of Sλ(V ) for all V or else all elements satisfying P come from simpler spaces Sμ(V ) for finitely many tuples μ λ. When λ consists of one partition, the second case in the theorem says that elements satisfying P have bounded strength in the following sense. Main Theorem I is an extension (in characteristic zero) of [20, Theorem 1.9] for homogeneous polynomials, which is the case where λ is a single partition with a single row. Example When λ = (d) d, the space Sλ,∞ consists of infinite degree-d forms in variables x1, x2,. Main Theorem II (Theorem 2.9.1) Let λ be a tuple of partitions, all of the same integer d ≥ 1.
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