AI 中文总结
研究赋值向量空间半线性关系的扎兰凯维奇界，给出超图边数上界等，证明相对扩展定理，通过量词消去得可定义关系界，还构造了特定复杂度且边数一定的不含\(K_{t,\ldots,t}\)的半线性点 - 盒图。
AI 中文摘要
我们为赋值向量空间中的半线性关系建立了绝对和相对的几乎线性扎兰凯维奇界。对于每个固定的元数和描述复杂度，一个不含\(K_{t,\ldots,t}\)的半线性\(r\)部超图最多有\[ O\!\left(n^{r - 1}(\log n)^c\right) \]条边，其中\(c\)仅取决于元数和赋值文字的数量。在二分图情况下，一个单独的任意迹论证给出了描述复杂度为\((\rho,s)\)时的显式界\(O(n(\log n)^{2s})\)。我们还证明了一个相对扩展定理：通过\(s\)次仿射移动半径比较将任何关系与具有遗传几乎线性轮廓的关系相交，对数指数最多增加\(2s\)。对于\(\mathbb{Q}_p\)和\(\mathbb{C}_p\)上的加法仿射赋值结构，量词消去将这些半线性结果转化为所有可定义关系的界。最后，在每个具有无限值群的赋值域上，我们构造了具有\(\Omega(n\log n/\log\log n)\)条边且描述复杂度为\((1,4)\)的不含\(K_{2,2}\)的半线性点 - 盒图。
英文摘要
We establish absolute and relative almost-linear Zarankiewicz bounds for semilinear relations in valued vector spaces. For every fixed arity and description complexity, a $K_{t,\ldots,t}$-free semilinear $r$-partite hypergraph has at most \[ O\!\left(n^{r-1}(\log n)^c\right) \] edges, where $c$ depends only on the arity and the number of valuative literals. In the bipartite case a separate arbitrary-trace argument gives the explicit bound $O(n(\log n)^{2s})$ for description complexity $(ρ,s)$. We also prove a relative extension theorem: intersecting any relation with a hereditary almost-linear profile by $s$ affine moving-radius comparisons increases the logarithmic exponent by at most $2s$. For the additive affine-valuative structures on $\mathbb Q_p$ and $\mathbb C_p$, quantifier elimination converts these semilinear results into bounds for all definable relations. Finally, over every valued field with infinite value group, we construct $K_{2,2}$-free semilinear point--box graphs of description complexity $(1,4)$ with $Ω(n\log n/\log\log n)$ edges.
Comments24 pages
