arXiv subjects
Publications and source records attributed to Zipei Nie.
In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a $(-2,3,2s+1)$-pretzel knot ($s\ge 3$) with slope $\frac{p}{q}$ is not left orderable if $\frac{p}{q}\ge 2s+3$, and that it is left orderable if $\frac{p}{q}$ is in a neighborhood of zero depending on $s$.
For a set $S$ of $d$ points in the $n$-dimensional projective space over a field of characteristic zero, we prove that the polynomials of degree $d$ whose zero sets are cones over $S$ do not span the vector space of polynomials of degree $d$ vanishing on $S$, if $d$ is odd and $d\ge 3$. Furthermore, they span a subspace of codimension at least two, if $n=2$, $d=1\pmod 4$ and $d\ge 5$.
We provide an efficient algorithm to compute the minimum area of a homotopy between two closed plane curves, given that they divide the plane into finite number of regions. For any positive real number $\varepsilon>0$, we construct a closed plane curve $γ$ such that the minimum area of a null homotopy of $2\cdotγ$ is less than $\varepsilon$ times that of $γ$. We also establish a lower bound on how complex a desired closed curve has to be.
The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state the theorem is via a sort of bounded degree Zariski closure operation: given a set, we consider all polynomials of some bounded degree vanishing on that set, and then the common zeros of these polynomials. For example, the degree d closure of d + 1 points on a line will contain the whole line, as any polynomial of degree at most d vanishing on the d + 1 points must vanish on the line. Our result is a bound on the size of a finite degree closure of a given set. Finally, we adapt our use of Hilbert functions to the method of multiplicities.