AI 中文总结
研究代数环面中光滑完全交的 amoeba 轮廓实度的通用上界，通过引入对数余法框架扩展 Pfaffian 方法，给出不同情况下的轮廓度估计，还利用伯恩斯坦定理获稀疏混合体积界，提供理论新解释。
AI 中文摘要
我们建立了代数环面中光滑完全交的 amoeba 轮廓的实度的首个通用上界。我们的方法通过引入基于对数余法丛、对数格拉斯曼映射和行列式秩条件的对数余法框架，将 Lang--Shapiro--Shustin 的 Pfaffian 方法从超曲面扩展到任意余维。我们证明，在合适的对数余法图表上，临界轨迹由对数余法矩阵产生的舒尔补方程局部定义。这在\(n\ge 2r\)和\(n<2r\)两种情况下都给出了明确的通用轮廓度估计。我们进一步用伯恩斯坦定理取代总次数论证，以获得由变换方程的牛顿多面体确定的稀疏混合体积界。这些估计通常比相应的通用界更精确，并提供了 Lang--Shapiro--Shustin 理论的高余维类似物以及通过对数余法几何对 amoeba 轮廓的新几何解释。
英文摘要
We establish the first universal upper bounds for the real degree of the contour of the amoeba of a smooth complete intersection in the algebraic torus. Our approach extends the Pfaffian method of Lang--Shapiro--Shustin from hypersurfaces to arbitrary codimension by introducing a logarithmic conormal framework based on the logarithmic conormal bundle, the logarithmic Grassmann map, and determinantal rank conditions. We prove that, on suitable logarithmic conormal charts, the critical locus is locally defined by Schur--complement equations arising from the logarithmic conormal matrix. This yields explicit universal contour-degree estimates in both regimes $n\ge 2r$ and $n<2r$. We further replace total-degree arguments by Bernstein's theorem to obtain sparse mixed-volume bounds determined by the Newton polytopes of the transformed equations. These estimates are frequently much sharper than the corresponding universal bounds and provide a higher-codimensional analogue of the Lang--Shapiro--Shustin theory together with a new geometric interpretation of amoeba contours through logarithmic conormal geometry.
Comments25 pages, all comments are welcome
