通过Steiner多项式在 tame 集上严格单调性数值
Strictly Monotone Numerosity on Tame Sets via the Steiner Polynomial
AI总结：
本文通过Steiner多项式定义了严格单调的内在体积多项式，连接了tame几何与非标准分析，并证明了其在数值理论中的应用。
摘要：
本文受积分几何启发，将 tame 几何与非标准分析连接。我们从Steiner多项式中省略二项式系数，定义出内在体积多项式Φ，该估值定义在o-最小结构中的有界可定义集上。我们证明使用这种归一化可使点集在码域ℝ[t]中通过终端行为解释时得到严格单调的估值。这导致了Hadwiger定理的代数版本：Φ是从ℂ(ℝ^∞)到ℝ[t]的唯一共正规连续、相似等价的环同态（除缩放外）。注意到严格单调性在数值理论（非标准分析的一个分支）中被镜像反映，我们证明了存在一种数值，其异常逼近内在体积多项式。这表明了不同领域之间的联系，使各方能够相互补充。
英文摘要：
This paper uses inspiration from Integral Geometry to connect Tame Geometry with Nonstandard Analysis. We omit binomial coefficients from the Steiner polynomial to define the \textit{intrinsic volume polynomial} $Φ$, a valuation defined on bounded definable sets in an o-minimal structure. We prove that using this normalization gives a strictly monotone valuation on point sets when the codomain $\mathbb{R}[t]$ is interpreted with ordering by end behavior. This leads to an algebraic version of Hadwiger's Theorem: $Φ$ is the unique conormal continuous, similarity-equivariant homomorphism of ordered rings from $\mathcal{C}(\mathbb{R}^\infty) \to \mathbb{R}[t]$ (up to scaling). Noting that strict monotonicity is mirrored in numerosity theory (a branch of nonstandard analysis), we prove existence for a numerosity that exceptionally approximates the intrinsic volume polynomial. This suggests a connection between disparate fields, allowing each to complement the other.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
