AI 中文总结
该研究分离出迹反射条件，证明其可传递渐近性，确定最小渐近分母，还将其应用于有限域上多项式图的纯关联结构，证明其构成完整k维渐近类且分母k最小。
AI 中文摘要
我们研究有限维渐近性何时能通过一致解释和可定义商集进行传递。环境计数产生有限多个渐近选项，但对应的参数胞腔可能无法在解释后的语言中被定义；一般而言，只能得到弱渐近性。我们分离出“迹反射”这一单侧下降条件，它可消除该阻碍。对于带有可定义选择子的解释，迹反射以显式分母界传递渐近性。对于一般可定义商集，不变渐近轮廓取代可定义选择，得到商集传递定理。我们确定了一个内在可见分母，证明其是最小的渐近分母，还给出了互素见证准则以判断依赖解释的界是否为最优。我们同时建立了公式层面、句法层面和语义层面的下降准则，证明了复合定理，并表明迹反射严格弱于一致弱双解释性。一个带轮廓的论证消除了一致有限域重构引入的非典范坐标参数。作为应用，我们研究射影商迹，证明对每个固定的k≥2，有限域上次数小于k的多项式图所产生的纯关联结构构成一个完整的k维渐近类；分母k是最小的，且所有成员均不含K_{k,n}（对所有n≥2）。
英文摘要
We study when finite-dimensional asymptoticity transfers through uniform interpretations and definable quotients. Ambient counting yields finitely many asymptotic alternatives, but the corresponding parameter cells may fail to be definable in the interpreted language; in general, one obtains only weak asymptoticity. We isolate "trace reflection", a one-sided descent condition that removes this obstruction. For interpretations with definable selectors, trace reflection transfers asymptoticity with explicit denominator bounds. For general definable quotients, invariant asymptotic profiles replace definable choice and yield a quotient-transfer theorem. We identify an intrinsic visible denominator and prove that it is the least possible asymptotic denominator, with a coprime-witness criterion for when the interpretation-dependent bound is sharp. We also establish formula-wise, syntactic, and semantic descent criteria, prove composition theorems, and show that trace reflection is strictly weaker than uniform weak bi-interpretability. A framed-profile argument removes the noncanonical coordinate parameters introduced by uniform finite-field reconstruction. As applications, we study projective quotient traces and prove that, for every fixed $k\geq2$, the pure incidence structures arising from graphs of polynomials of degree less than $k$ over finite fields form a full $k$-dimensional asymptotic class. The denominator $k$ is minimal, and every member is $K_{k,n}$-free for all $n\geq2$.
Comments55 pages
