In 1979 Valiant introduced the complexity class VNP of p-definable families of polynomials, he defined the reduction notion known as p-projection and he proved that the permanent polynomial and the Hamiltonian cycle polynomial are VNP-complete under p-projections. In 2001 Mulmuley and Sohoni (and independently B\"urgisser) introduced the notion of border complexity to the study of the algebraic complexity of polynomials. In this algebraic machine model, instead of insisting on exact computation, approximations are allowed. This gives VNP the structure of a topological space. In this short note we study the set VNPC of VNP-complete polynomials. We show that the complement VNP \ VNPC lies dense in VNP. Quite surprisingly, we also prove that VNPC lies dense in VNP. We prove analogous statements for the complexity classes VF, VBP, and VP. The density of VNP \ VNPC holds for several different reduction notions: p-projections, border p-projections, c-reductions, and border c-reductions. We compare the relationship of the VNP-completeness notion under these reductions and separate most of the corresponding sets. Border reduction notions were introduced by Bringmann, Ikenmeyer, and Zuiddam (JACM 2018). Our paper is the first structured study of border reduction notions.

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关于VNP完整性和边界复杂性的说明

1979年，Valiant引入了p可定义的多项式族的复杂度VNP，他定义了归约概念p投影，他证明了p投影下的永久多项式和哈密顿循环多项式是VNP完全的。在2001年，Mulmuley和Sohoni（以及独立的B'urgisser）将边界复杂度的概念引入多项式代数复杂度的研究中，在这种代数机器模型中，不是坚持精确的计算，而是允许近似值。在此简短的笔记中，我们研究了VNP完全多项式的集合VNPC，证明补码VNP \ VNPC在VNP中是密集的，令人惊讶的是，我们还证明了VNPC在VNP中是密集的。复杂度级别VF，VBP和VP的语句。VNP \ VNPC的密度适用于几种不同的归约概念：p投影，边界p投影，c约简和边界c约简。我们比较了这些减少下的VNP完整性概念的关系，并分离了大多数相应的集合。减少边界的概念由Bringmann，Ikenmeyer和Zuiddam提出（JACM 2018）。我们的论文是第一个结构化的边界缩减概念研究。