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arXiv · 2505.16323

Characterization of polynomials by their invariance properties

J. M. Amira·Ya-Qing Hu
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Abstract

We prove that certain classical groups $G\subseteq {\rm GL}(d,\mathbb{R}^d)$ serve to characterize ordinary polynomials in $d$ real variables as elements of finite-dimensional subspaces of $C(\mathbb{R}^d)$ that are invariant by changes of variables induced by translations and elements of $G$. We also show that, if the field $\mathbb{K}$ has characteristic $0$, the elements of $\mathbb{K}[x_1,\cdots,x_d]$ admit a similar characterization for $G= {\rm GL}(d,\mathbb{K})$.

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J. M. Amira, Ya-Qing Hu. 2025-05-22. Characterization of polynomials by their invariance properties. https://arxiv.org/abs/2505.16323

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