不适用于良好回归的多项式映射及其应用
Polynomial maps which are not good for nice recurrence and applications
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中文总结 AI 辅助
本文针对二元有限域上的可数无限维向量空间，构造了不适用于良好回归的多项式映射，否定了Bergelson和McCutcheon的相关猜想，还建立了多项式回归与密度多项式Hales-Jewett猜想的关联。
中文摘要 AI 辅助
设𝔽₂为二元有限域，𝔽₂^ω表示𝔽₂上的可数无限维向量空间。我们证明，与在零点消失的多项式映射p:ℤ→ℤ总是适用于良好回归的情况不同，存在满足p(0_{𝔽₂^ω})=0_{𝔽₂^ω}的多项式映射p:𝔽₂^ω→𝔽₂^ω，它们不具备该性质。这否定了Bergelson和McCutcheon在约2000年提出的猜想，该猜想预测对于任意可数无限阿贝尔群H和G，每个满足p(0_H)=0_G的多项式映射p:H→G都适用于良好回归。此外，我们提出一种动力学机制，表明沿多项式路径的交集大小具有次数敏感性（即使仅考虑弱混合系统时也是如此）。我们还证明，对于在零点消失的次数至多为d的𝔽₂^ω值多项式的Furstenberg-Sarkozy定理，等价于密度多项式Hales-Jewett猜想的d维对称差弱化形式。因此，正如本文详细解释的那样，我们的发现不仅为多项式回归现象提供了新的视角，还为证明或否定密度多项式Hales-Jewett猜想的可能策略提供了约束。
英文摘要
Let $\mathbb F_2$ be the finite field with two elements and let $\mathbb F_2^ω$ denote the countably infinite-dimensional vector space over $\mathbb F_2$. We show that, unlike the case of polynomial maps $p:\mathbb Z\rightarrow\mathbb Z$ vanishing at zero which are always good for nice recurrence, there are polynomials $p:\mathbb F_2^ω\rightarrow \mathbb F_2^ω$ with $p(0_{\mathbb F_2^ω})=0_{\mathbb F_2^ω}$ which fail to have this property. This disproves a conjecture of Bergelson and McCutcheon (c. 2000), which predicted that for any countably infinite abelian groups $H$ and $G$, every polynomial map $p:H\to G$ with $p(0_H)=0_G$ is good for nice recurrence. Moreover, we develop a dynamical mechanism which shows that the magnitude of intersections along polynomial paths is degree-sensitive (even when one considers only weakly mixing systems). Among other things, we also show that the Furstenberg-Sarkozy theorem for $\mathbb F_2^ω$-valued polynomials of degree at most $d$ vanishing at zero is equivalent to a $d$-dimensional symmetric-difference weakening of the density polynomial Hales-Jewett conjecture. Thus, as we explain in detail in this paper, our observations not only shed new light on the phenomenon of polynomial recurrence but also constrain possible strategies for proving or disproving the density polynomial Hales-Jewett conjecture.
