十点齐次插值

Homogeneous Interpolation and Some Continued Fractions

We prove: if $d/m < 2280/721$, there is no curve of degree $d$ passing through $n = 10$ general points with multiplicity $m$ in $\bf{P}^2$. Similar results are given for other special values of $n$. Our bounds can be naturally written as certain palindromic continued fractions.

喷流插值

Interpolation on Jets

我们表明，对于任何有限的一般联盟对$(x_i,L_i)_i$，其中$x_i\in L_i$是一个点的线$L_i$在射影$n$ -空间，因子$m_ix_i$的$L_i$有最大的秩相对于齐次$d$形式的所有$d\geq 0$和所有$m_i\geq 0$模预期的数值限制。

Emptiness of homogeneous linear systems with ten general base points

In this paper we give a new proof of the fact that for all pairs of positive integers (d, m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty.

Interpolation and The Weak Lefschetz Property

Our starting point is a basic problem in Hermite interpolation theory, namely determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem if the number of points is small compared to the dimension of their linear span. This also allows us to establish results on the Hilbert function of ideals generated by powers of linear forms. The Verlinde formula determines such a Hilbert function in a specific instance. We complement this result and also determine the Castelnuovo-Mumford regularity of the corresp…

关于部分多项式插值

On partial polynomial interpolation

本文将Alexander-Hirschowitz定理推广到包含在一般双点并集中的任意零维方案，在多项式插值框架下证明了除d=2的五个例外情况外，给定任意多个一般一阶偏导线性组合值的次数≤d的多项式仿射空间具有预期维数。