2018/08/28 by David Kazhdan, Tamar Ziegler, Kazhdan, David +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1808.09439
openalex publication_date 2018/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a field, V a k-vector space and X be a subset of V . A function f:X→ k is weakly polynomial of degree ≤ a, if the restriction of f on any affine subspace L⊂ X is a polynomial of degree ≤ a. In this paper we consider the case when X= \mathbb X (k) where \mathbb X is a complete intersection of bounded codimension defined by a high rank polynomials of degrees d, char(k)=0 or char (k)>d and either k is algebraically closed, or k=\mathbb F q,q>ad. We show that under these assumptions any k-valued weakly polynomial function of degree ≤ a on X is a restriction of a polynomial of degree ≤ a on V. Our proof is based on Theorem 1.11 on fibers of polynomial morphisms P:\mathbb F qn→ \mathbb F qm of high rank. This result is of an independent interest. For example it immediately implies a strengthening of the result of [4].