Characterization of polynomials by their invariance properties

2025/05/22 by J. M. Amira, Amira, J. M., Ya-Qing Hu +1
Mathematics · #39A70 #39B05 #39B22 #39B52 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2505.16323

openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that certain classical groups G⊆ \rm GL(d,ℝ^d) serve to characterize ordinary polynomials in d real variables as elements of finite-dimensional subspaces of C(ℝ^d) that are invariant by changes of variables induced by translations and elements of G. We also show that, if the field \mathbbK has characteristic 0, the elements of \mathbbK[x₁,⋯,x_d] admit a similar characterization for G= \rm GL(d,\mathbbK).

Citations

• Using Aichinger's equation to characterize polynomial functions
• Separately polynomial functions
• On Loewner's characterization of polynomials
• On Montel and Montel-Popoviciu theorems in several variables

Related

• On invariant fields of vectors and covectors
• A group action on multivariate polynomials over finite fields
• Invariant theory of a special group action on irreducible polynomials over finite fields
• Polynomial invariants of classical subgroups of GL₂: Conjugation over finite fields
• Generic orbits, normal bases, and generation degree for fields of rational invariants