论文标题
品种关节
Joints of varieties
论文作者
论文摘要
我们将guth-katz关节定理从线到品种概括。一个特例说，$ n $ plaes（2 flats）的6个维度（在任何字段上）具有$ o（n^{3/2}）$ intims，其中关节是这些平面三倍的关节，并非全部都躺在某些超平面中。更一般而言，当$ n $ n $ n $ n $ n $的一组二维代数品种取代一组$ n $ n $ n $ $ n $的二维代数变种时，我们证明了相同的界限，而关节是定期的三个品种，这些品种的切线并非全部包含在某些超平面中。对于具有多种任意维度（称为Carbery的猜想）的多种乘积的关节，我们最普遍的结果给出了上限，直至恒定因素。我们的主要创新是一种将多项式方法扩展到更高维对物体的新方法，将多项式的程度及其在多种多样的一组点上消失的程度及其消失的顺序。
We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of $N$ planes is replaced by a set of 2-dimensional algebraic varieties of total degree $N$, and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety.
