论文标题
总和和单射线曲线
Sumsets and monomial projective curves
论文作者
论文摘要
本说明的目的是利用添加剂组合学与单射线曲线的几何形状之间的新关系。我们将一组有限的非阴性整数$ a = \ {a_1，\ cdots，a_n \} $ a单个投影曲线$ C_A \ subset \ subset \ mathbb p^{n-1} _ {k} _ {k} $ sa：= \ {a_ {a_ {i_1}+\ cdots+a_ {i_s} \ mid 1 \ le i_1 \ le \ cdots \ cdots \ le i_s \ le i_s \ le n \ le n \} $ cluent。 $ c_a $的奇异性决定了$ | sa | $的渐近行为，等效地，$ c_a $的希尔伯特多项式和$ sa $的渐近结构。我们表明，某些添加剂的逆问题可以转化为希尔伯特多项式的刚度，并通过使用加法组合学的结果来改善Castelnuovo-Mumford单射线曲线的上限。
The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers $A=\{a_1,\cdots, a_n\}$ a monomial projective curve $C_A\subset \mathbb P^{n-1}_{k}$ such that the Hilbert function of $C_A$ and the cardinalities of $sA:=\{a_{i_1}+\cdots+a_{i_s}\mid 1\le i_1\le \cdots \le i_s\le n\}$ agree. The singularities of $C_A$ determines the asymptotic behaviour of $|sA|$, equivalently the Hilbert polynomial of $C_A$, and the asymptotic structure of $sA$. We show that some additive inverse problems can be translate to the rigidity of Hilbert polynomials and we improve an upper bound of the Castelnuovo-Mumford regularity of monomial projective curves by using results of additive combinatorics.
