Geometric postulation of a smooth function and the number of rational points
Geometric postulation of a smooth function and the number of rational points
复制标题
光滑函数的几何假设和有理点的数量
DOI：
--
复制
发表时间：
1991
期刊：
影响因子：
--
通讯作者：
J. Pila
中科院分区：
文献类型：
--
作者：
J. Pila
本文致力于改进和推广Bombieri[1]的一些结果，并得到函数图上积分格点个数的上界。考虑图为Γ的充分光滑函数f(X)，且d为正整数.本文的主要目的是考虑Γ上不在d次实代数曲线上的积分点.文[1]的主要引理表明，这些点不能相对于函数的某些范数太近.我们在这里追求相对于主要引理的两个不同的目标。第一个是得到函数f(X)上控制Γ与任意d次代数曲线相交的重数的局部条件。这实质上是对主要引理的假设的研究，构成了标题的几何假设。事实上，在第2节和第3节中，我们将得到实代数曲线的任何线性空间的这样的条件。第五节给出了整点的应用。例如，如果f(X)∈C在[0，1]上且W(f，2)=f‘’|f‘3f’‘0 f IV 4f’‘6f’‘f 5f IV 20f’‘|不是零，则对每个e>；0，
This paper is devoted to giving refinements and extensions of some of the results of Bombieri and the author [1] obtaining upper bounds for the number of integral lattice points on the graphs of functions. Consider a sufficiently smooth function f(x) with graph Γ, and a positive integer d. The main device of that paper was to consider integral points on Γ that do not lie on any real algebraic curve of degree d. The Main Lemma of [1] shows that such points cannot be too close together relative to certain norms of the function. We pursue here two different goals relative to the Main Lemma. The first is to obtain local conditions on the function f(x) that control the multiplicity of the intersection of Γ with any algebraic curve of degree d. This is essentially an investigation into the hypotheses of the Main Lemma, and constitutes the Geometric Postulation of the title. Indeed, in sections 2 and 3 we will obtain such conditions for any linear space of real algebraic curves. The applications to integral points are given in section 5. For example, we show that if f(x) ∈ C on [0, 1] and W (f, 2) = f ′′ ∣∣∣∣∣ f ′′′ 3f ′′ 0 f iv 4f ′′′ 6f ′′ f 5f iv 20f ′′′ ∣∣∣∣∣ is nowhere zero then, for every e > 0,